ELEMENTS 


OF 


MECHANICAL 
DRAWING 

ALFRED  A.TITSWORTH 


LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


Class 


ELEMENTS  OF  MECHANICAL  DRAWING 

IN     TWO    PAR  TS 
PART   I.     FOR   BEGINNERS 

EXERCISES   IN  THE   USE   OF   INSTRUMENTS   AND   USE   OF   SCALES,    AND   SIMPLE    PROBLEMS   IN 

PROJECTIONS,.   INTERSECTIONS,    AND   DEVELOPMENTS 

PART    II 

PROBLEMS   IN    DESCRIPTIVE   GEOMETRY 


BY 

ALFRED   A.    TITSWORTH,    M.    Sc.,    C.E 

Professor  of  Civil  Engineering  and  Graphics  in  Rutgers  College 


FIRST  EDITION.       FIRST  THOUSAND 


NEW   YORK 

JOHN   WILEY    &   SONS 

LONDON  :    CHAPMAN    &    HALL,    LIMITED 

1906 


Copyright,   1906 

BY 
ALFRED   A.   TITSWORTH 


ROBERT  DRUMMOND,   PRINTER,   NEW  YORK 


PREFACE. 


THE  aim  of  this  treatise  is  to  supply  a  course  in  Me- 
chanical Drawing  which  shall  constitute  a  satisfactory 
foundation  for  all  advanced  courses. 

Part  I  provides  a  course  for  beginners,  to  acquaint 
them  with  the  use  of  drafting  instruments  and  the  prin- 
ciples of  projections.  The  instruction  is  detailed  and 
full  to  the  extent  that  this  course  may  be  pursued  not 
only  in  colleges  and  technical  schools,  but  also  in  second- 
ary schools,  and  even  by  the  earnest  student  without  a 
teacher. 

Part  II  comprises  problems  in  Descriptive  Geometry  to 
be  constructed  in  the  drafting-room  to  supplement  class- 
room instruction. 


The  third-angle  projection  has  been  adopted  in  both 
parts  in  conformity  with  the  general  practice  in  the 
drafting  departments  of  business  and  engineering  con- 
cerns where  practical  work  is  done. 

These  courses  have  been  taught  in  Rutgers  College 
with  excellent  results,  and  they  are  believed  to  supply  a 
satisfactory  basis  for  the  higher-courses. 

I  wish  to  acknowledge  my  indebtedness  to  Mr.  Richard 
Morris,  Associate  Professor  of  Mathematics  and  Graphics 
in  Rutgers  College,  for  timely  and  valuable  suggestions. 
j-  ALFRED  A.  TITSWORTH. 


RUTGERi  COIXEGE,   NEW  BRUNSWICK,  N.  J., 

September,  1906. 


iii 


161913 


TABLE  OF  CONTENTS. 


PART  I. 
CHAPTER  I. 

PAGE 

DRAWING  INSTRUMENTS  AND  THEIR  USES 

List  of  Materials  Required.     Care  and  Use  of  Instruments. 

CHAPTER  II. 

GENERAL  INSTRUCTIONS  AND  PRACTICE  EXERCISES  WITH  INSTRU- 
MENTS         7 

Diagram  for  Dividing  the  Plates.  Plain  and  Gothic  Letters 
and  Numerals.  Plate  of  Letters.  Directions  for  Drawing 
Lines.  Exercises  for  Practice  with  the  Ruling-pen.  Exer- 
cises for  Drawing  Arcs  and  Circles  with  the  Compasses.  Ex- 
ercises in  Drawing  Simple  Designs,  Straight  Lines  Tangent 
to  Curves,  and  in  the  Use  of  the  Irregular  Curve. 

CHAPTER  III. 

GEOMETRIC  CONSTRUCTIONS  AND  CONSTRUCTION  AND  USE  OF 

SCALES 27 

Geometric  Problems.  Construction  and  Use  of  Scales — 
Problems. 

CHAPTER  IV. 

PROJECTIONS 38 

Definitions.     Conventions.     Shade-lines.    The  Cube.     Sec- 


tions. Rectangular  Prism.  Supplemental  Projections. 
Hexagonal  Prism.  Development.  Intersections.  Oblique 
Projections.  The  Pyramid.  Section  and  Development  of 
the  Pyramid.  Hexagonal  Pyramid.  Intersection  of  Pyramid 
and  Prism.  The  Cylinder.  Section  and  Development  of 
the  Cylinder.  Intersection  of  Prism  and  Cylinder.  Inter- 
section of  Two  Cylinders.  Intersection  of  the  Cone  and 
Prism.  Intersection  of  the  Cone  and  Oblique  Cylinder. 

CHAPTER  V. 

ISOMETRIC  AND  OBLIQUE  PROJECTIONS,  SHADES,  SHADOWS,  AND 

PERSPECTIVE 82 

Isometric  Projection.  Shade-lines.  Oblique  Projections. 
Cavalier  Projection.  Pseudo -perspective.  Shade-lines. 
Shades  and  Shadows.  Perspective.  Vanishing-points. 
Problems. 

PART  II. 

PROBLEMS  IN  DESCRIPTIVE  GEOMETRY. 

Instructions.  Conventions.  Problems  on  the  Point,  Line, 
and  Plane.  Shade  Lines.  Problems  on  Tangent  Surfaces. 
Intersections  and  Developments.  Shades  and  Shadows. 
Problems  in  Shades  and  Shadows.  Perspective.  To  Find 
the  Vanishing-point  of  a  System  of  Parallel  Lines  Analytically 
and  Graphically.  Problems  in  Perspective in 


ERRATA. 

Page  i,  2cl  column,  nth  line,  for  inch  read  inches. 

Page  i,  2d  column,  131)1  line,  for  foot  read  inch. 

Page  10,  2d  column,  8th  line,  for  letter  read  letters. 

Page  18,  ist  columsi,  2d  line,  omit  the  words  and  end. 

Page  20,  ist  column,  7th  and  8th  lines,  for  the  lengths  of  the  radii  of  the  circles  are  equal  to  a  half  and 
whole  space  respectively  read  the  radius  of  the  larger  circles  is  equal  to  a  space,  the  smaller  circles  are  drawn 
tangent  to  them. 

Page  27,  ist  column  4th  line,  fcr  top  read  bottom. 

Page  31,  2d  column,  loth  line,  for  if"  read  ij". 

Page  34,  ist  column   4th  line,  for  -fg"  read  A"- 

Page  34,  2d  column,  i8th  line,  for  scale  of  2  read  scale  of  3. 

Page  36,  ist  column,  4th  line,  for  J"  read  J". 

Page  58,  ist  column,  last  line,  add  In  the  projection  on  V,  C»,  is  at  (4^",  3!")- 

Page  58,  2d  column,  4th  and  5th  lines  omit. 

Page  64,  2d  column,  4  h  line,  add  V  is  at  (11.5",  4.25"). 

Page  74,  ist  column,  22d  line,  for  4.5"  read  4". 

Page  84,  ist  column,  5th  line,  for  oy  read  oz. 

Page  86,  2d  column,  4th  line,  for  less  read  greater. 

Page  89,  in  figure  (6),  for  a  read  d. 

Page  92,  ad  column  I5th  line,  for  EK  read  Ej,'. 

Page  94,  ist  column,  8th  line,  add  The  edge  of  the  cube  is  2". 

Page  94,  2d  column,  4th  line,  add  The  height  of  the  cylinder  is  2$",  and  the  radius  of  the  base  is  i". 

Page  112,  2d  column,  ist  line,  for  AB  read  HV. 

Page  121,  in  figure  i,  height  of  drawing  table,  for  3'  4"  read  3'  3". 

Page  123,  2d  column,  8th  line,  for  4\"  read  4$". 

Page  123,  2d  column,  gth  line,  for  f '  read  }". 


OF  THE 

t     UNIVERSITY 


MECHANICAL    DRAWING. 


PART    I. 


CHAPTER  I. 


DRAWING    INSTRUMENTS  AND   THEIR   USES. 


1.  Mechanical  Drawing,  or  Drawing  with  Instruments, 
demands  exactness  as  to  the  direction,  form,  and  mag- 
nitude of  lines,  in  contradistinction  to  free-hand  drawing 
in  which  these  are  estimated  or  determined  by  the  eye. 
The  object  of  the  exercises  as  far  as  Plate  10  is  to  familiar- 
ize the  student  with  the  uses  of  drafting  instruments, 
and  to  teach  him  to  be  exact  and  careful  in  performing 
the  several  operations  with  them. 

2.  List  of  Materials  Required : 

Compasses,  5|  inches,  having  one  fixed  leg  with  a 
needle-point,  the  other  leg  having  interchangeable  pen 
and  pencil  points  and  a  lengthening-bar; 

Hair-spring  Dividers,  5  inches; 

Spring  Bow-spacers,  about  3  inches; 

Spring  Bow-pencil,  about  3  inches; 

Spring  Bow-pen,  about  3  inches; 


Ruling-pen,  unhinged,  5  inches,  best  make. 

The  instruments  thus  far  mentioned  may  be  purchased 
in  a  case  together. 

Drawing-board,  20  by  26  inches; 

T  Square,  24-inch  blade; 

30-6o-degree  Triangle,  the  longest  side  n  inches. 

45-degree  Triangle,  the  side  9  inches; 

Scroll,  or  Irregular  Curve  (similar  to  Keuffel  &  Esser 
Co.'s  No.  26,  hard  rubber); 

Boxwood  Flat  Scale,  12  inches,  graduated  in  i,  -J,  £, 
and  \  inch  to  the  foot; 

Boxwood  Triangular  Scale,  12  inches,  graduated  in  10, 
20,  30,  40,  50,  and  60  parts  to  the  foot ; 

Boxwood  Rectangular  Protractor,  6  by  if  inches,  with 
diagonal  scales  and  scale  of  chords; 

Two  Drawing-pencils,  one  3H,  the  other  6H; 


MECHANICAL    DRAWING. 


Pencil-pointer  (sandpaper  pad,  or  a  sheet  of  oo  sand- 
paper); 

Ink  and  Pencil  Erasers; 

One  dozen  Thumb-tacks,  or  a  package  of  1-02.  Copper 
Tacks,  for  fastening  the  paper  to  the  drawing-board; 

Bottle  of  Black  Water-proof  Drawing-ink; 

Bottle  of  Red  Water-proof  Drawing-ink; 

One  Ball-pointed  Pen  (D.  Leonardt  &  Co.); 

Two  Writing-pens  (Gillott's  Nos.  303  and  290  respect- 
ively); 

Pen-holder; 

Horn  Center;  and 

Good  Drawing-paper  as  needed. 

3.  Care  and  Uses  of  Instruments. — The  instruments 
should  be  kept  free  from  dried  ink  and  dust,  by  wiping 
on  a  piece  of  chamois  skin  or  lintless  cloth.  Ink  should 
never  be  allowed  to  dry  in  the  pen.  The  nibs  of  the  pen 
should  be  well  separated  when  not  in  use,  and  never 
screwed  together  close  enough  to  cause  pressure. 

Be  careful  never  to  use  the  edges  of  the  T  square  or 
triangles  as  a  guide  for  the  knife  to  cut  the  paper,  as  the 
edges  are  thus  often  cut  into  and  ruined  for  use  as  straight- 
edges. A  large  pair  of  shears  are  most  convenient  for 
cutting  the  sheets  of  paper  into  proper  sizes.  The  use 
of  a  knife  roughens  the  surface  of  the  drawing-board. 

The  instruments  should  be  of  good  quality,  although 
the  costliest  are  not  essential  for  good  work.  But  with 
imperfect  instruments  accurate  drawings  cannot  be  made. 


4.  The  Compasses. — These  are  used  for  describing  arcs 
of  circles,  and  should  always  be  held  so  that  the  jointed 
legs  are  nearly  perpendicular  to  the  surface  of  the  paper, 
the  fingers  lightly  clasping  the  joint,  and  the  compasses 
slightly  inclined  in  the  moving  direction.  One  leg  has  a 
fixed  needle-point,  and  to  the  other  leg  may  be  fitted 
either  a  pencil-joint  or  a  pen-point;  and  when  arcs  of 
large  radii  are  to  be  drawn  a  lengthening-bar  may  be 
used  to  extend  this  leg.  There  are  joints  to  both  legs 
of  the  compasses,  so  that  the  lower  part  may  be  held 
perpendicular  to  the  surface  of  the  paper.  (This  is  done 
in  order  that  the  center  will  not  change  in  describing  the 
arc,  and  also  that  both  nibs  of  the  pen  will  touch  the 
paper  and  thus  avoid  making  a  ragged  line.)  The  needle 
is  smoothly  pointed  at  one  end,  and  on  the  other  there  is 
a  shoulder;  the  shoulder  prevents  the  point  from  enter- 
ing the -paper  too  far.  Just  sufficient  pressure  should  be 
put  upon  the  compasses  to  keep  the  point  from  slipping 
and  still  not  cause  it  to  penetrate  the  paper.  The  pencil 
in  the  compasses  should  be  sharpened  by  wearing  away 
the  lead  on  the  outside  with  the  sandpaper,  so  that  it 
shall  taper  towards  the  side  next  to  the  other  leg  of  the 
compasses,  leaving  that  side  straight  to  bring  it  in  close 
contact  with  the  other  leg. 

5.  Hair-spring  Dividers.  —  These  are  used  for  setting 
off  measurements  on  the  drawing,  but  are  never  used  to 
describe  arcs;  there  is,  therefore,  no  necessity  for  joints 
in  the  legs.  A  set-screw  is  adjusted  to  one  leg  so  that  a 


DRAWING   INSTRUMENTS  AND   THEIR   USES. 


3 


slow  motion  may  be  given  to  the  movement  of  one  leg 
to  more  accurately  obtain  the  measurement  to  be  set  off 
on  the  drawing. 

6.  The  Spring  Bow-spacers,  Spring  Bow-pen,  and  Spring 
Bow-pencil. — These  are  used,  the  spacers  to  space  small, 
equal  divisions  of  a  line,  and  the  pen  and  pencil  to  draw 
very  small  arcs  of  circles.     They  will  be  found  very  con- 
venient and  often  necessary. 

7.  Ruling-pen.  —  This   is   used   for  drawing   ink-lines, 
either    straight  or  curved.      It  is  always  guided  by  a 
straight-edge,  as  that  of  a  T  square  or  side  of  a  triangle 
when  drawing  straight  lines,  or  a  curved  edge,  as  that  of 
a  scroll,  when  drawing  curved  lines.     The  pen  should  be 
held  perpendicular  to  the  surface  of  the  paper,  except  that 
it  should  be  inclined  slightly,  say  5  degrees,  in  the  moving 
direction.     The  nibs  of  the  pen  should  always  be  held 
so  that  they  are  parallel  to  the  edge,  either  straight  or 
curved,  against  which  it  is  held  very  lightly.     The  pres- 
sure against  the  edge  should  be  so  slight  as  to  avoid 
shutting  the  nibs  together  and  thus  checking  the  flow  of 
ink.     Pens  become  dulled  from  frequent  use  and  need 
sharpening  in  order  that  fine  and  even  lines  may  be 
drawn  with  them;    but  to  be  successful  in  the  delicate 
operation    of    sharpening    the    nibs   a    magnifying-glass 
should  be  used  to  watch  the  process  of  wearing  away  the 
the  surface  to  a  sharp  edge  on  a  stone  or  emery-cloth. 
The  finishing  touches  must  be  made  on  an  Arkansas 
stone  of  fine  grain,  or  upon  emery  polishing-cloth  (or 


paper),  or  one  may  learn  from  the  instrument-maker  the 
proper  way  to  do  it.  It  is  well  to  have  on  hand  two 
ruling-pens,  so  that  one  may  always  be  in  condition  for 
ruling  fine  lines,  and  also  that  when  different  grades  of 
lines  are  required  in  the  same  drawing  one  pen  may  be 
set  for  one  grade  of  line  and  the  other  for  the  other  grade. 

8.  The  Drawing-board. — Upon  this  the  paper  is  fast- 
ened with  the   thumb-tacks  or  i-oz.  copper  tacks.     It 
should  have  a  plane,  smooth  surface.     The  edges  should 
be  straight,  so  that  when  the  head  of  the  T  square  slides 
along  and  against  the  edge  of  the  board  the  blade  of  the 
T  square  will  move  in  parallel  positions.     Only  one  edge 
of  the  board  should  be  used  with  the  T  square  during  the 
process  of  any  one  drawing;    and  the  board  should  be 
so  turned  as  to  bring  that  edge  on  the  left  of  the  drafts- 
man. 

9.  T  Square    and    Triangles. — While    the  blade  of  the 
T  square  is  used  as  a  guide  to  the  pencil  or  pen  to  draw 
parallel  lines  which  are  horizontal,  that  is,  parallel  to  the 
top  edge  of  the  drawing-board,  a  triangle  may  be  made 
to  rest  against  the  blade  of  the  T  square,  and  by  sliding 
it  along  the  blade  the  other  side  may  be  used  as  a  guide 
for  drawing  parallel  lines  perpendicular  to  the  edge  of 
the  T  square.     To  draw  vertical  lines,   therefore,  do  not 
use  the  T  square  against  the  top  or  bottom  edge  of  the  board, 
but  use  one  of  the  triangles  resting  against  the  blade  of  the 
T  square  while  the  head  slides  against  the  left  edge  of  the 
board.     This   is   important,   because   the   edges   of   the 


MECHANICAL  DRAWING. 


drawing-board  may  not  be,  and  cannot  be  kept,  perpen- 
dicular to  each  other.  To  draw  lines  making  angles  of 
30,  45,  and  60  degrees  with  the  edge  of  the  T  square, 
with  one  side  of  the  proper  triangle  resting  against  the 
blade  of  the  T  square,  use  the  hypothenuse  as  a  guide 
for  the  pencil.  By  using  the  two  triangles  together 
angles  of  15  and  75  degrees  may  be  laid  off.  Another 
and  important  use  which  may  be  made  of  the  triangles 
is  to  draw  parallel  lines  on  the  paper  in  any  direction, 
and  also  lines  perpendicular  to  each  other  in  any  direc- 
tion. Thus,  if  the  hypothenuse  of  one  triangle  be  made 
to  coincide  with  a  given  line,  and  an  edge  of  the  other 
triangle  (or  any  straight-edge)  be  placed  against  one  of 
the  sides  of  the  first  triangle,  and  the  first  triangle  be 
made  to  slide  along  the  second,  the  hypothenuse  will 
move  in  positions  parallel  to  the  given  line.  By  changing 
the  straight-edge  to  the  other  side  of  the  first  triangle 
the  same  system  of  parallel  lines  may  be  transferred  to 
another  part  of  the  paper.  To  draw  lines  perpendicular 
to  this  system  of  parallel  lines,  put  the  other  side  of  the 
first  triangle  against  the  straight-edge  and  the  hypothe- 
nuse will  then  be  in  a  position  perpendicular  to  its  first 
direction.  The  triangles  should  be  thus  used  by  the 
student  to  practise  drawing  lines  parallel  and  perpen- 
dicular to  each  other  until  he  is  familiar  with  the  manipu- 
lations. 

10.  Scroll,   or  Irregular  Curve. — This  is  used  to  direct 
the  inking  of  curved  lines  which  are  not  arcs  of  circles. 


Some  part  of  the  edge  of  the  scroll  can  usually  be  found 
to  coincide  with  the  curved  line  on  the  paper  which 
should  be  already  drawn  in  pencil,  and  after  allowing  a 
sufficient  space  between  the  edge  of  the  scroll  and  the 
line  for  the  pen  to  move  over  the  line  while  resting  against 
the  scroll,  this  part  is  inked  in;  the  scroll  is  then  moved 
until  some  other  part  is  found  to  coincide  with  a  con- 
tinuous part  of  the  pencilled  line  and  that  part  also  inked; 
continuing  thus  the  pencilled  line  is  completely  inked. 
Great  care  should  be  observed  to  turn  the  pen  as  it 
moves  along  the  edge  of  the  scroll  so  that  the  nibs  shall 
remain  tangent  to  the  edge,  otherwise  the  ink  will  run 
out  of  the  pen  on  the  scroll  and  produce  a  blot  on  the 
paper. 

u.  Boxwood  Flat  Scale. — This  scale  is  graduated  on 
the  edges  by  lines  that  are  one  inch  apart  in  one  case, 
and  different  fractions  of  an  inch  in  other  cases,  and  the 
space  at  one  end  of  the  graduation  is  divided  into  12 
equal  parts.  For  example,  in  one  case  the  space  is  one- 
half  an  inch  laid  off  successively  along  the  edge,  and  the 
end  space  is  divided  as  stated  above.  If  the  scale  of  the^ 
drawing  is  \  an  inch  to  i  foot,  that  is,  one  foot  on  the 
object  to  be  represented  is  made  \  inch  on  the  drawing, 
then  this  scale  may  be  used  directly  to  scale  off  distances 
on  the  drawing  representing  the  corresponding  feet  and 
inches  on  the  object. 

12.  The   Triangular  Scale. — This  is  a  scale   of   equal 
parts,  that  is.  the  six  different  faces  are  divided  so  that 


DRAWING   INSTRUMENTS    AND   THEIR   USES. 


on  one  and  another  there  are  10,  20,  30,  40,  50,  and  60 
divisions  to  each  inch,  respectively.  This  is  called  an 
engineer's  scale,  and  is  generally  used  where  long  dis- 
tances are  represented  by  proportionately  short  distances 
on  the  drawing.  For  example,  if  it  is  desired  to  represent 
on  the  drawing  by  one  inch  a  distance  of  60  feet,  the 
scale  of  60  parts  to  the  inch  would  be  used  where  each 
division  of  the  scale  would  represent  one  foot. 

13.  The    Protractor. — This  is  used  to  measure  angles 
between  lines  on  a  drawing,  and  also  to  lay  off  from  one 
line  at  a  given  point  an  angle  another  line  is  to  make 
with  the  first  line.     In  this  particular  form  of  protractor 
three  of  the  edges  are  divided  in.  degrees  from  zero  to 
180,  the  middle  of  the  fourth  side  being  the  center  of 
the  radial  lines  marking  the  degrees.     It  also  contains  a 
scale  of  chords  upon  which  are  shown  the  chords  of  each 
degree  from  zero  to  90  on  an  arc  whose  radius  is  the 
chord  of  60  degrees  on  the  scale.     This  scale  is   also 
used  to  measure  and  lay  out  angles.    On  the  reverse  side 
is  a  diagonal  scale  which  "divides  the  inch  (or  half-inch) 
into  tenths  and  hundredths.     The  construction  of  the 
diagonal  scale  and  a  scale  of  chords  are  exercises  to  be 
done  by  the  student  later,   when   further  explanation 
will  be  given.     , 

14.  Pencils. — The  3H   pencil  should  be  sharpened  on 
one  end  to  a  long,  sharp,  rounded  point  for  marking 
letters  and  figures.     The  6H  pencil  should  be  sharpened 
on  one  end  as  is  the  3H,  but  on  the  other  to  a  long,  flat, 


chisel-shaped  edge,  with  the  corners  slightly  rounded, 
thus: 


FIG.  i. 

In  drawing  a  line  through  a  point  the  straight-edge 
(T  square  or  triangle)  should  be  brought  to  half  cover 
the  point  and  the  pencil  held  vertical  with  the  flat 
side  of  the  chisel-edge  pressed  against  the  straight-edge; 
then  if  a  line  is  drawn  it  will  pass  through  the  point. 
If  the  round  point  is  used  instead,  allowance  must  be 
made  by  placing  the  straight-edge  some  distance  from 
the  point,  so  that  when  the  pencil  is  held  vertical  it  will 
pass  through  the  point,  otherwise  if  the  pencil-point  is 
inclined  toward  the  straight-edge  to  make  the  pencil 
pass  through  the  point,  it  is  likely  to  run  under  the  edge 
where  the  ruler  is  not  held  firmly  against  the  paper  and 
a  straight  line  is  not  drawn.  The  points  of  the  pencil 
should  be  kept  very  sharp  by  cutting  away  the  wood 
from  the  lead  and  wearing  the  lead  to  the  proper  shape 
on  the  sandpaper  pencil-pointer  and  afterwards  polishing 
the  lead  off  on  a  piece  of  cloth. 

15.  Ink. — The  best  ink  is  made  by  rubbing  the  China 
or  India  stick  ink  in  a  small  quantity  of  water,  on  a  por- 
celain slab,  until  a  sufficient  quantity  has  been  worn  off 
to  make  the  ink  of  the  desired  consistency  when  mixed 


6 


MECHANICAL   DRAWING. 


with  water.  This,  however,  requires  considerable  time, 
and,  except  for  very  fine  drawings,  it  is  hardly  worth 
while  when  there  are  inks  of  good  quality  on  the  market 
already  mixed  and  bottled.  The  best  of  these  answer  for 
all  ordinary  drawings.  None  of  the  writing-inks  should 
ever  be  used  in  the  pens  or  the  drawing-instruments,  as 
the  acids  in  these  inks  quickly  corrode  and  roughen  the 
edges  of  the  nibs,  making  the  pens  useless.  Colored  inks 
are  made  from  artist's  water-colors,  or  they  are  found  in 
the  market,  as  is  the  black  ink,  already  mixed  and  bottled. 
The  ordinary  colored  inks  made  for  writing  purpose  must 
be  avoided.  India  ink  rests  on  the  surface  of  the  paper 
and  may  easily  be  erased.  It  does  not  penetrate  the 
paper  as  do  the  chemical  inks. 


16.  The  Writing-pens.  —  These  are  for  making  letters 
and  figures  on  the  drawings. 

17.  Horn  Center. — This  is  a  small  disc  of  thin,  transpar- 
ent horn  with  three  fine,  sharp  points  to  hold  it  in  place 
on  the  paper,  and  is  used  to  place  over  the  center  of  sev- 
eral concentric  circles,  so  that  the  frequent  placing  of  the 
compass-point  on  the  center  will  not  make  a  large  hole  and 
destroy  the  good  appearance  and  accuracy  of  the  drawing. 

18.  Drawing-paper.  —  Whatman's    drawing-paper,    the 
hot-pressed,  which  has  a  smooth  surface,  or  any  other 
good  drawing-paper,  may  be  used  for  the  drawings  of 
this  course.     If  Whatman's  paper  is  used,  that  known  as 
"Medium  "  17"  by  22"  makes  two  plates  the  size  to  be 
used  in  this  course. 


CHAPTER   II. 


GENERAL   INSTRUCTIONS   AND   PRACTICE    EXERCISES   WITH  INSTRUMENTS. 


19.  The  series  of  drawings  following  are  to  be  made 
on  plates  of  uniform  size,  14  inches  long  by  9.5  inches 
wide.  A  border-line  extends  around  the  plate,  one-half 
an  inch  from  the  outside  edge,  except  on  the  left  end, 
where  it  is  one  inch  from  the  edge  to  allow  for  binding. 
The  dimensions  inside  of  the  border-line,  therefore,  will 
be  12.5  inches  by  8.5  inches.  Some  plates  will  be  sub- 
divided, in  pencil  only,  into  six  equal  squares,  with  sides 
3.5  inches,  separated  from  each  other  and  from  the 
border-line  by  a  space  of  one-half  an  inch,  as  shown  in 
Diagram  A.  The  squares  are  numbered  from  left  to 
right,  beginning  in  the  upper  row.  The  initial  point  or 
origin  of  a  square  is  the  lower,  left-hand  corner,  the 
reference  axes  being  the  lower  and  left  sides  of  the  square. 
x  represents  the  horizontal,  and  y  the  vertical,  ordinate 
of  any  point.  For  example,  if  the  point  is  2  inches  from 
the  left-hand  axis  and  1.5  inches  from  the  bottom  axis, 
the  point  is  designated  thus:  #  =  2",  y  =  i.5",  or  simply 
(2",  1.5"),  the  x  distance  being  always  placed  first. 
When  the  problem  or  drawing  requires  the  whole  plate 
the  squares  are  omitted  and  the  left-hand  and  bottom 
border-lines  will  be  the  reference  axes. 


The  word  Plate  and  the  number  of  the  plate  are  to  be 
printed  at  the  upper  right-hand  corner,  midway  between 
the  border-line  and  the  outside  edge,  and  the  studenfs 
name  in  the  lower  right-hand  corner  in  a  corresponding 
position.  The  height  of  the  letters  is  to  be  &  of  an 
inch,  no  part  of  the  printing  to  extend  to  the  right  of 
the  right-hand  border-line. 

20.  A  line  that  is  parallel  to  the  bottom   axis  of   the 
plate  will  be  known  as  a  horizontal  line,  or  simply  a  hori- 
zontal, and  a  perpendicular  to  the  horizontal  line  will  be 
known  as  a  vertical  line,  or  simply  a  vertical. 

21.  Plain,  or  Gothic,  Letters  and  Numerals. — The  correct 
formation  of   letters  and  their  proper  spacing  can  be 
learned    only   by  a  careful  study  of  details,  and  prac- 
tice alone  will  make  the  student  an  adept  at  lettering.* 
But  for  the  letters  to  be  made  in  the  following  series  of 
drawings  the  simple  style  of  letter  shown  in  Plate  A  will 
be  used. 

A  few  general  principles  will  be  noted  in  regard  to 
this  style  of  letter. 

*  See  "Lettering,"  by  C.  E.  Sherman,  and  "Lettering  for 
Draftsmen,  Engineers,  and  Students,"  by  C.  W.  Reinhardt. 

8 


fENERAL    INSTRUCTIONS    AND    PRACTICE    EXERCISES    WITH   INSTRUMENTS 

DIAGRAM  A. 


•f 

J_ 


CO 


10 


MECHANICAL   DRAWING. 


a.  Capital  letters   and   numerals   which   go   together 
should  be  the  same  height,  and  the  small  letter  (lower 
case)  corresponding  should  be  three- fifths  the  height  of  the 
capital. 

b.  The  breadth  of  letters  in  proportion  to  their  height 
varies  but  a  good  proportion  is  as  follows:   Calling  one- 
fifth  of  the  height  of  the  capital  letter  a  unit,  then  the 
breadth  of 


A,C,G,K,0,Q,X,Y=4f  units, 
D.T.V.Z  -4*      " 

=4       " 
=3i      " 


B,  K,  R,  S 
F,  H,  N,  P,  U 
L 

J 
M 
W 
I 


=  31 
-3t 
=5 
=6* 


and 


=  a  single  line. 


c.  For  the  small  letters  the  breadth  of  the  body  part 
should  be  a  shade  narrower  than  its  height,  that  is,  a 
shade  less  than  three  units,  and  for  those  letters  which 
extend    above  or  below    the    body  part    the  extension 
is  two  units  except  in  the  t,  which  extends  above  one 
unit  only. 

d.  In  the  inclined  letters  the  slant  is  3  horizontal  to 
8  vertical.     The  length  of  the  stems  is  the  same  as  in 
the  upright  letters,  making  the  vertical  height  a  little 


less.  The  breadth  of  inclined  letters  is  governed  by  the 
rule  which  applies  to  upright  letters. 

e.  Spacing. — The  general  principle  to  be  observed  in 
spacing  is  to  make  the  letters  of  a  word  appear  to  be  a 
uniform  distance  apart.  If  the  distances  of  the  letters 
apart  are  equal,  the  effect  will  be,  in  general,  to  make 
them  appear  to  the  eye  unequally  separated,  because 
when  the  letter  are  placed  side  by  side  their  various 
forms  in  different  combinations  give  varying  areas  of 
space.  To  illustrate:  In  the  word  WAVERLY,  if  the 

letters  are  equally  spaced,  thus,  VVAV  E^RLlY,  they 
do  not  appear  to  be,  but  when  unequally  spaced  thus, 
VW\VERI_y  -  tne  spacing  is  correct.  Again,  in  the 
word  INMATE,  if  equally  spaced  thus,  JJ^J^/\[TE, 
the  appearance  to  the  eye  is  not  symmetrical,  but  when 
printed  thus,  |[\|[\/1/\T  I_>  ^e  sPacmg  being  unequal, 
still  the  appearance  is  correct.  A  little  examination  of 
these  and  other  examples  will  lead  to  the  general  conclu- 
sion that  if  the  areas  between  the  letters  appear  to  the 
eye  to  be  as  nearly  equal  as  possible  the  spacing  will  be 
correct.  Where  two  letters  with  straight,  vertical  stems, 
as  the  I  and  N,  or  the  N  and  M,  in  the  word  INMATE 
come  together,  the  distance  must  be  increased,  and  when 
such  letters  as  the  L  and  Y,  with  inclined  stems,  come 
together,  as  in  the  word  WAVERLY,  the  distance  must 
be  lessened.  It  would  be  impracticable  to  attempt  to 


GENERAL   INSTRUCTIONS    AND   PRACTICE   EXERCISES   WITH   INSTRUMENTS. 


11 


make  rules  governing  the  multitude  of  permutations 
and  combinations  of  letters  that  are  likely  to  come 
together  in  printing  words,  and  so,  aside  from  the 
general  conclusion  reached  above,  the  student  must 
depend  upon  his  eye  to  determine  the  proper  spacing 
of  letters. 

/.  Two  horizontal  guide-lines  should  always  be  drawn , 
in  pencil,  to  mark  the  tops  and  bottoms  of  the  letters. 
Letters  with  curved  stems,  as  C,  G,  O,  Q,  S,  should 
extend  a  shade  above  and  below  the  guide-lines;  while 
letters  bounded  by  horizontal  lines  at  top  and  bottom, 
or  either,  as  B,  D,  E,  F,  L,  should  be  a  little  scant,  if 
anything,  in  order  that  the  two  kinds  of  letters  coming 
together  may  appear  to  be  of  the  same  height.  Letters 
terminating  in  vertical  or  slant  lines  at  top  or  bottom,  or 
both,  as  F,  H,  I,  K,  etc.,  or  in  a  sharp  angle,  as  A,  M,  W, 
etc.,  should  extend  a  shade  beyond  the  guide-line  for  the 
same  reason.  Letters  B,  E,  H,  S,  and  Z  should  be  made 
larger  at  the  bottom  than  at  the  top  in  order  that  the 
symmetrical  parts  should  appear  to  be  equal.  If  the  stu- 
dent will  invert  such  letters,  when  correctly  printed,  he 
will  observe  how  much  the  bottom  is  larger  than  the  top. 

22.  Copy  the  letters  of  Plate  A,  after  reading  carefully 
Art.  21. 

The  lower  guide-line  of  the  upper  row  is  at  y=6.g 
inches. 

The  lower  guide-line  of  the  second  row  is  at  y  =  5-75 
inches. 


The  lower  guide-line  of  the  third  row  is  at  y  =  4-8 
inches. 

There  should  be  four  guide-lines  for  the  small  letters: 
one  guide-line  to  mark  the  bottom  of  the  body  part  of 
the  letter,  a  line  three  units  above  to  mark  the  height  of 
the  body  part,  a  line  two  units  below,  and  another  line 
five  units  above;  the  highest  and  lowest  lines  marking 
the  extremities  of  such  letters  as  b  and  p  above  and 
below  the  body  part. 

The  lower  guide-line  of  the  fourth  row  is  at  y=3-7 
inches. 

The  lower  guide-line  of  the  fifth  row  is  at  y  =  2.6 
inches. 

The  lower  guide-line  of  the  sixth  row  is  at  y  =  i.6 
inches. 

The  height  of  the  capital  letters  and  the  numerals  is 
three-sixteenths  of  an  inch;  the  unit  is,  therefore,  one- 
fifth  of  this  distance.  The  height  of  the  body  part  of 
the  small  letters  is  three  units.  With  the  spring  bow- 
spacers  divide  three-sixteenths  of  an  inch  into  five 
equal  spaces,  then  with  this  space  unchanged  on  the 
instrument  it  may  be  used  to  measure  the  widths  of 
the  letters  as  given  in  Art.  21.  The  vertical  height  of 
the  slant  capitals  and  numerals  is  obtained  by  laying 
off  on  the  proper  slant  five  units  and  drawing  the  upper 
guide-line  through  the  extremity  of  the  distance  thus 
laid  off.  Similarly  the  guide-lines  for  the  small  letters 
may  be  obtained. 


12 


MECHANICAL  DRAWING. 


23.  Directions. — (a)  Use  the  ball-point  pen  for  drawing 
the  capital  letters  and  numerals,  and  the  303  Gillott  pen 
for  the  small  letters. 

b.  Learn  to  make  the  letters  free-hand,  that  is,  without 
the  use  of  the  ruling-pen  and  straight-edge,  and  with  a 
steady,  single  stroke,  without  sketching. 

c.  In  drawing  straight  stems  the  stroke  should  be  a 
pulling  stroke,  and  never  a  pushing  one.     To  draw  the 
straight  horizontal  strokes,  turn  the  drawing-board  or 
change  your  position  in  order  to  draw  the  pen  toward  you. 

d.  In  drawing  curved   letters,  as  the  0  for    instance, 
make  it  in  two  strokes,  joining  them  at  top  and  bottom, 

thus:    ^Q)  .     The  arrows  show  the  direction  in  which  the 

pen  should  move.  The  illustration  shows  a  separation 
at  top  and  bottom  to  indicate  where  each  stroke  is  to 
begin  and  end;  the  student  shcfuld  join  these  carefully 
in  his  work. 

e.  To  make  the  curved  letters  symmetrical  they  should 
be  "blocked  in,"  that  is,  they  should  be  circumscribed 
by  a  rectangle  for  the  upright  letters,  and  by  a  parallelo- 
gram for  the  slant  letters,  and  the  curves  made  tangent 
to  the  sides. 

/.  In  drawing  the  straight,  horizontal,  and  vertical 
strokes  take  great  pains  to  make  them  respectively 
exactly  horizor  tal  and  vertical.  A  very  slight  variation 
of  the  top  of  a  vertical  stroke  to  the  right  makes  it  look 
tipped,  while  a  slight  variation  the  other  way  is  not 


noticeable.     In  fact  a  slight  inclination  of  the  top  to  the 
left  is  to  be  recommended. 

24,  Directions  about  Drawing  Lines. — To  draw  hori- 
zontal lines,  the  edge  of  the  T  square  guides  the  motion 
of  the  pencil  and  ruling-pen,  and  to  draw  vertical  lines 
the  side  of  a  triangle,  when  the  other  side  rests  against 
the  edge  of  the  T  square.  Lines  making  angles  of  30, 
45,  and  60  degrees  with  a  horizontal  are  drawn  by 


FIG.  2. 

placing  one  side  of  the  proper  triangle  against  the  T 
square  and  using  the  hypothenuse  as  a  guide  for  the 
pencil.  To  draw  lines  making  angles  of  15  and  75 
degrees  with  a  horizontal,  the  triangles  are  placed  as 
in  Fig.  2. 

If  the  line  is  to  make  15  or  75  degrees  with  the  hori- 
zontal on  the  other  side,  interchange  the  positions  at  the 
angles  A  and  B  of  the  outer  triangle. 


PLATE  A. 


ABCDEFGHIJKLMNOPQRSTUVWXYZ 

1234567890 

abcdef  g  hij  k  I  mn  opqrstuvwxyz 
ABC  DEFGHIJKL  MNOP  QFfS  TUVWXYZ 

1234567890 

abcdef qhijklmn  opqr  sfu  vwxyz 


13 


14 


MECHANICAL   DRAWING. 


To  draw  a  line  or  a  series  of  lines  -parallel  or  perpendic- 
ular to  any  oblique  line,  the  triangles  may  be  used  as  ex- 
plained in  Art.  10. 

25.  Lines  will  be  thus  designated: 


Full  lines   (unbroken" 
or  continuous) " 

Broken  lines  (dashes)   " 


Dotted 


Broken  and  dotted 


Lines  may  be  heavy,  medium,  or  light  in  their  relations  to 
each  other.  What  may  be  considered  a  light  line  in  one 
drawing  as  compared  with  other  lines  in  the  same  draw- 


ing, in  another  drawing  may  be  considered  medium  or 
heavy.  The  lines  below  are  light,  medium,  and  heavy, 
respectively,  with  relation  to  each  other. 


Care  should  be  taken  to  make  all  heavy,  medium,  and 
light  lines  of  uniform  grade  or  thickness,  respectively,  on 
the  same  drawing. 

"  Broken  lines"  should  have  the  dashes  the  same 
length,  about  \"  long,  and  the  spaces  equal,  about  one- 
third  the  length  of  the  dash. 

Dotted  lines  are  made  by  holding  the  drawing-pen 
vertical  and  merely  touching  the  paper.  If  the  pen  is 
properly  sharpened  the  mark  made  on  the  paper  will 
then  be  a  slightly  elongated  dot.  The  open  space  and 
the  dot  should  be  equal  in  length. 


16 


MECHANICAL    DRAWING. 


PLATE  1. 


26.  Exercise  for  Practice  in  the  Use  of  the  Ruling-pen. 

— Lay  out  the  lines  of  the  plate  in  pencil  as  in  Diagram 
A,  after  fastening  the  paper  to  the  drawing-board  with 
the  thumb-tacks  or  i-oz.  copper  tacks.  Measure  dis- 
tances consecutively  from  one  line  with  the  boxwood 
flat  scale.  Use  the  T  square  as  a  guide  for  the  pencil  in 
drawing  horizontal  lines,  and  one  of  the  triangles  for  ver- 
tical lines. 

In  squares  i,  3,  and  5,  with  the  spring  bow-spacers, 
by  trial,  divide  the  left  side  of  each  into  eight  equal 
spaces,  adjusting  the  screw  of  the  spacers  until  the 
points  are  the  exact  distance  apart.  Then  with  the 
chisel-edge  of  the  pencil  draw  horizontal  lines  through  the 
points  of  division  from  the  left  to  the  right  sides  of  the 
square.  Draw  very  light  lines  with  the  pencil,  as  all  pen- 
cilled lines  are  to  be  either  covered  by  ink  or  erased. 

In  square  2  divide  the  top  side  of  the  square  into  eight 
equal  spaces,  and  draw  lines  through  the  points  of  divi- 
sion, using  the  side  of  the  triangle  as  a  guide  for  the 
pencil,  while  the  other  side  of  the  triangle  slides  along 
the  edge  of  the  T  square. 

In  squares  4  and  6  draw  lightly,  in  pencil  only,  a  diag- 
onal of  each  square  perpendicular  to  the  direction  in 
which  the  lines  are  to  be  drawn,  then  divide  each  diagonal 


into  ten  equal  parts,  and  through  these  points  of  division, 
with  the  hypothenuse  of  the  45-degree  triangle  as  a 
guide,  draw  lines  limited  by  the  sides  of  the  square. 

After  drawing  all  these  lines  in  pencil  they  are  to  be 
drawn  in  ink.  Place  ink  between  the  nibs  of  the  ruling- 
pen  by  means  of  the  ink-filler  in  the  stopper  of  the  bottle, 
or  with  an  ordinary  writing-pen,  so  that  there  will  be 
some  pressure  of  the  ink  towards  the  point,  but  not 
enough  to  cause  the  ink  to  drop  out  by  a  slight  jerk  of 
the  hand.  Have  a  waste  piece  of  paper  to  try  the  pen  on, 
and  adj  ust  the  set-screw  in  the  pen  to  the  proper  width 
to  produce  a  line  of  the  required  thickness.  Follow  the 
copy  in  Plate  I  as  to  the  kinds  of  lines  to  be  drawn. 
Observe  directions  in  Art.  7  as  to  the  use  of  the  ruling- 
pen,  and  draw  the  lines  in  ink  over  the  pencil-lines, 
starting  exactly  on  the  line  of  the  side  of  the  square, 
moving  the  pen  to  the  right  and  stopping  exactly  on 
the  opposite  side.  In  the  horizontal  lines  move  the  pen 
from  left  to  right,  in  vertical  lines  from  bottom  to  top, 
in  the  diagonal  lines  of  square  4  from  the  bottom  tow- 
ards the  top,  and  in  square  6  from  the  top  towards  the 
bottom. 

Draw  the  border-lines  and  print  the  word  PLATE  and 
its  number  and  your  name  as  directed  in  Art.  19. 


/  / 


/  /  / 

/    /    /  / 
/   /  / 


/  /  /  / 

/     / 


/  / 

/    /    / 


/ 


PLATE  1. 


\\\\ 

\     \     \ 


\ 


\ 
\     \ 


\ 


\ 


\ 


\ 


\ 


\ 


\ 


17 


18 


MECHANICAL   DRAWING. 


PLATE  2. 


27.  Exercise  in  Drawing  Long  Lines  with  the  Ruling- 
pen.— Draw  a  line,  in  pencil  only,  on  each  side  and  end 
of  the  plate,  one-half  an  inch  inside  of  the  border-line. 
Divide  the  line  on  the  left  side  into  twenty  equal  spaces, 
thus:  Use  the  hair-spring  dividers  to  bisect  the  distance 
between  the  upper  and  lower  lines,  and  to  bisect  the 
two  equal  spaces  thus  formed,  then  with  the  spring  bow- 
spacers  divide  each  of  the  four  remaining  spaces  into 
five  equal  spaces.  Draw  horizontal  lines  in  pencil 
through  each  of  the  points  of  division.  Afterwards 
draw  over  the  pencil-lines  ink-lines,  beginning  with 
the  upper  line  for  the  first  line,  and  ending  with  the 
bottom  line  for  the  twenty-first,  as  follows: 

Three  full,  light  lines  (see  Art.  25). 

Three  full,  medium  lines, 


Three  full,  heavy  lines, 

Three  broken,  light  lines, 

Three  broken,  medium  lines, 

Three  broken,  heavy  lines, 

One  dotted,  light  line, 

One  dotted,  medium  line,  and 

One  dotted,  heavy  line. 

Begin  exactly  on  the  pencil-line  one-half  an  inch  inside 
of  the  left-hand  border-line,  hold  the  pen  at  the  same  angle 
(see  Art.  7)  from  start  to  finish,  and  end  exactly  on  the 
pencil-line  one-half  an  inch  inside  the  right-hand  border- 
line. After  inking  all  lines  erase  with  a  soft  pencil- 
eraser  all  pencil-lines  which  show,  ink  in  the  border-lines, 
and  print  the  word  Plate,  its  number  and  your  name, 
as  before. 


GENERAL  INSTRUCTIONS   AND  PRACTICE   EXERCISES   WITH  INSTRUMENTS. 


19 


PLATE  3. 


28.  Exercise  in  Drawing  Arcs  and  Circles  with  the  Com- 
passes.— Divide  the  plate  into  six  equal  squares  as  in 
Diagram  A.  In  square  i,  find  the  center  by  drawing, 
in  pencil  only,  the  diagonals  of  the  square.  Through 
the  center  draw  a  horizontal  line  and  divide  it  into  six- 
teen equal  spaces,  or  one-half  of  it  into  eight.  Draw 
in  pencil  with  the  compasses  concentric  circles  passing 
through  the  points  of  division,  the  center  of  the  square 
being  the  common  center  of  the  circles.  Use  the  spring 
bow-pencil  for  the  two  smaller  circles.  In  order  to  avoid 
gouging  a  hole  in  the  paper  in  drawing  several  circles 
with  the  same  center,  use  the  horn  center  and  let  the 
needle-point  of  the  compasses  rest  on  it.  (Observe  care- 
fully the  instructions  in  Art.  5.) 

In  square  2,  find  the  enter  of  the  square  as  before 
and  divide  the  lower  half  of  the  vertical  line  through 
the  center  into  eight  equal  parts,  and  through,  the 
points  of  division  draw  circles  whose  radii  vary  in 
length  consecutively  from  one  to  eight  spaces.  The 
centers  of  the  respective  circles  fall  on  the  points  of 
division,  consecutively,  from  bottom  to  middle.  Be 
sure  that  all  the  circles  are  exactly  tangent  to  each  other 
at  the  bottom. 


In  square  3,  draw  a  horizontal  line  through  the  center 
of  the  square  and  divide  it  into  sixteen  equal  spaces. 
With  these  points  of  division  as  centers,  consecutively, 
and  radii  varying  in  length  from  one  space  to  eight  spaces, 
draw  semicircles,  beginning  at  the  left,  below  the  hori- 
zontal line,  then  with  radii  varying  consecutively  from 
eight  spaces  to  one  space  draw  the  remaining  eight 
semicircles  above  the  horizontal  line.  Be  careful  that 
the  circles  are  exactly  tangent  where  they  meet  on  the  hori- 
zontal line,  and  also  on  the  outside. 

In  square  4,  find  the  center  of  the  square  and  draw  a 
circle  tangent  to  the  four  sides  of  the  square.  Draw 
horizontal  and  vertical  lines  through  the  center,  and  the 
other  radial  lines,  using  the  30-6o-degree  triangle.  By 
trial  find  a  circle,  its  center  on  the  vertical  line,  which 
shall  be  tangent  to  the  outside  circle  and  to  the  two 
adjacent  radial  lines,  or  this  may  be  done  by  inscribing  a 
circle  in  the  equilateral  triangle  formed  by  the  two 
adjacent  radial  lines  and  the  side  of  the  square.  Through 
this  center  on  the  vertical  line  draw  a  circle  whose  center 
is  at  the  center  of  the  square.  This  circle  will  cut  all 
the  other  radial  lines  in  the  centers  of  the  corresponding 
circles.  With  these  centers  draw  the  smaller  circles,  as 


20 


MECHANICAL    DRAWING. 


shown  in  the  plate,  tangent  to  the  outside  circle  and  to 
each  other. 

In  square  5,  divide  the  left  and  bottom  sides  of  the 
square  each  into  four  equal  spaces  and  draw,  in  pencil 
only,  horizontal  and  vertical  lines  through  the  points  of 
division.  The  intersections  of  these  lines  are  the  centers 
of  the  respective  circles.  The  lengths  of  the  radii  of  the 
circles  are  equal  to  a  half  and  a  whole  space  respectively. 
Be  very  careful  to  determine  the  exact  positions  of  the 
centers  of  the  circles,  so  that  the  circles  will  be  tangent 
and  not  intersect. 

In  square  6,  find  the  center  of  the  square  and  draw 
a  circle  tangent  to  the  sides.  Draw  another  circle  with 
the  radius  one-sixteenth  of  an  inch  less.  Draw  hori- 
zontal and  vertical  lines  through  the  center  of  the  square, 
and  the  other  radial  lines  by  using  the  45-degree  triangle. 


Draw  a  circle  with  a  radius  one  and  one-sixteenth 
inches,  its  center  at  the  center  of  the  square,  cutting 
the  radial  lines  in  the  centers  of  the  smaller  circles,  re- 
spectively, which  are  drawn  tangent  to  the  inner  of  the 
two  outside  circles.  Three  other  concentric  circles  are 
then  drawn  with  radii  of  one-half,  seven-eighths,  and 
fifteen-sixteenths  of  an  inch  respectively. 

After  these  exercises  are  done  in  pencil  they  are  then 
to  be  drawn  in  ink.  Follow  the  copy  in  the  thickness  of 
the  lines.  Use  the  compasses  for  the  larger  circles  and 
the  spring  bow-pen  for  the  smaller.  (Observe  carefully 
the  instructions  in  Art.  5.)  Use  the  horn  center  where 
several  concentric  circles  are  to  be  drawn  to  avoid  in- 
accuracy caused  by  the  needle-point  of  the  compasses 
wearing  a  hole  in  the  paper  at  the  center,  and  allowing 
the  point  to  shift  its  position  as  the  circle  is  drawn. 


n 


4- 


PLATE  3. 


22 


MECHANICAL    DRAWING. 


PLATE  4. 


29.  Exercise  in  Drawing  Simple  Designs,  Straight  Lines 
Tangent  to  Curves,  and  in  the  Use  of  the  Irregular  Curve.— 
In  square  r,  divide  two  opposite  sides  into  eight  equal 
spaces  and  through  the  point  of  division  nearest  one 
corner,  and  each  alternate  point,  draw,  first  in  pencil 
and  afterwards  in  ink,  one  set  of  parallel  lines.  After 
drawing  lines  through  the  alternate  points  on  one  side 
continue  through  the  corresponding  points  on  the  oppo- 
site side.  Or  one  side  may  be  divided,  as  stated,  with 
the  bow-spacers  and  the  corresponding  points  found  on 
the  opposite  side  by  transferring  them  across  by  the 
use  of  the  T  square. 

Use  the  45-degree  triangle,  one  side  resting  against 
the  edge  of  the  T  square,  and  the  hypothenuse  as  a  guide 
for  the  pencil.  Reverse  the  triangle  to  draw  parallel, 
diagonal  lines  drawn  in  the  other  direction. 

In  square  2,  divide  the  sides  as  before  and  draw 
parallel,  diagonal  lines  through  each  point  of  division. 
Ink  in  also  the  heavier  horizontal  and  vertical  lines 
as  shown. 

In  square  3,  divide  two  adjacent  sides  into  eight  equal 
spaces,  and  through  the  points  of  division  draw,  in  pencil 


only,  the  respective  horizontal  and  vertical  lines  (Art. 
9)  as  shown  in  Fig.  3.  The  diagonal  lines,  making 
an  angle  of  45  degrees  with  a  horizontal,  are  to  be 
drawn  as  shown  in  the  figure,  first  in  pencil,  then  in 
ink. 

In  square  4,  draw  the  horizontal  and  vertical  lines,  in 
pencil  only,  as  in  Fig.  3.     The  small  circles  in  Fig.  4 


FIG.  3. 


FIG. 


enclose  the  centers  of  the  several  circles  or  arcs  which 
are  shown.  Draw  the  arcs  first  and  afterwards  the 
straight  lines  tangent  to  the  arcs.  These  should  be  drawn 
in  pencil  first,  then  in  ink. 


GENERAL   INSTRUCTIONS   AND   PRACTICE    EXERCISES    WITH   INSTRUMENTS. 


23 


In  square  5,  divide  two  adjacent  sides  of  the  square 
into  sixteen  equal  spaces.  Draw,  in  pencil  only,  hori- 
zontal and  vertical  lines  through  the  points  of  division  as 
shown  in  Fig.  5.  The  smallest  circles  mark  the  centers 


FIG.  5. 


FIG.  6. 


of  the  respective  circles,  which  have  radii  equal  in  length 
to  one  space  and  two  spaces  respectively.  A  study  of 
Fig.  5  will  make  clear  which  lines  are  to  be  inked  and 
which  erased.  Draw  all  arcs  first  and  the  straight  lines 
tangent  to  the  arcs  afterwards. 

In  square  6,  draw  horizontal  and  vertical  lines  as  in 
Fig.  5.  Study  carefully  Fig.  6. 

At  A,  Fig.  6,  the  points  a,  b,  b',  c,  c',  d,  d',  e,  and  e' 
are  located  as  follows:  e  is  at  the  intersection  of  the 
fct  vertical  line  from  the  right  side  with  the  fourth 
horizontal  line  from  the  top  side;  d  is  on  the  fifth  hori- 
zontal line  and  bisects  the  distance  between  the  first  and 


second  vertical  lines;  c  is  on  the  sixth  horizontal  line 
and  bisects  the  remaining  distance,  or  is  three-fourths 
the  distance  from  the  first  to  the  second  vertical  line; 
b  is  on  the  seventh  horizontal  line  and  bisects  the  remain- 
ing distance,  or  is  seven-eighths  the  distance  from  the 
first  to  the  second  vertical  line;  and  a  is  at  the  inter- 
section of  the  eighth  horizontal  with  the  second  vertical 
line,  e',  d',  c',  b'  have  corresponding  positions,  respect- 
ively, on  the  other  side  of  the  middle,  or  eighth,  hori- 
zontal line.  Corresponding  points  are  found  on  all  four 
sides  of  the  square.  Through  these  points,  with  the 
irregular  curve  as  a  guide  to  the  pencil,  a  continuous  line 
is  drawn.  If  no  part  of  the  edge  of  the  irregular  curve 
will  coincide  with  all  the  points  at  the  same  time,  then 
different  parts  of  the  curve  must  be  made  to  coincide 
with  at  least  three  points  at  the  same  time,  and  care 
must  be  exercised  to  have  the  ending  of  one  part  of  the 
curved  line  and  the  beginning  of  another  part  continuous 
and  tangent  to  the  same  straight  line  at  the  point  of 
junction  (see  Art.  10).  At  B  the  next  step  is  shown. 
Small  circles  are  drawn  with  radii  equal  to  one-half  a 
space,  tangent  to  the  irregular  curve  at  the  points  a,  b,  e, 
etc.,  that  is,  the  radii  are  respectively  normal  to  the 
curve  at  these  points.  At  C  the  next  step  is  shown. 
The  inner  curve  is  drawn,  with  the  irregular  curve  as  a 
guide  to  the  pencil,  tangent  to  these  small  circles. 

The  arcs  at  the  corners  and  sides  of  the  square,  with 
their  respective  centers  and  lengths  of  radii  of  one  or 


24 


MECHANICAL   DRAWING. 


more  spaces,  as  is  shown  in  the  figure,  are  drawn  as 
indicated.  Straight  lines  tangent  to  each  of  the  two 
curves  which  they  join  are  drawn  as  shown  at  D  and  E. 
The  design  in  the  middle  of  the  square  is  made  up  of  arcs 
of  circles  whose  centers  are  shown  in  the  figure.  The 
lengths  of  the  radii  are  always  one  or  more  spaces,  except 
at  F  and  the  three  other  corresponding  positions,  where 


the  center  comes  at  the 
a  vertical  line,  and  the 
to  make  the  arcs  tangent 
The   arcs  are  drawn 
points  as  shown  in  the 
done  great  care  should  be 
line  exactly  at  the  tangent 


intersection  of  a  horizontal  and 
length  of  the  radius  is  such  as 
on  the  line  joining  their  centers, 
in  pencil  beyond  the  tangent 
figure,  .but  when  the  inking  is 
taken  to  end  each  arc  or  straight 
point  of  junction. 


PLATE  4. 


X 

X 


X 
X 


XJ 

X 


25 


CHAPTER  III. 


GEOMETRIC    CONSTRUCTIONS,    AND    CONSTRUCTION   AND    USE    OF    SCALES. 


30.  Geometric  Constructions.  —  In  constructing  the 
following  problems  the  number  of  the  problem  is  to  be 
printed  at  the  top  of  the  square,  midway  between  the 
sides,  the  top  of  the  numeral  coinciding  with  the  upper 
line  of  the  square,  and  its  height  to  be  three-sixteenths 
of  an  inch. 

All  given  lines  in  the  problems  are  to  be  drawn  light, 
full,  in  black  ink;  required  lines,  medium,  full,  in  black 
ink;  construction  lines,  light,  full,  in  red  ink;  and  explana- 
tion and  projecting  lines,  light,  dotted  in  red  ink.  (Lines 
as  shown  in  the  accompanying  plates  when  broken  rep- 
resent construction  lines,  and  when  dotted  represent 


explanation  or  projecting  lines.)  When  a  point  is  given 
draw  a  very  small  circle  around  it  in  red  ink,  and  when 
required  the  same  in  black  ink. 

Foot  and  feet  will  be  represented  by  one  accent,  and 
inch  and  inches  by  two  accents;  thus,  12  feet  8  inches  is 
represented  by  12'  8". 

The  problems  are  to  be  constructed  by  the  use  of  the 
compasses,  a  straight-edge,  and  scales.  Nevertheless  when 
either  horizontal,  vertical,  or  oblique  lines  are  given  lines 
the  T  square  and  triangles  may  be  used  in  drawing  them. 

The  problems  are  to  be  constructed  within  the  squares 
in  consecutive  order. 


27 


28 


MECHANICAL    DRAWING. 


PLATE  5. 


Prob.    i.  Bisect  a  given  horizontal  line  zj"  long  by  a  required  vertical 
line  2"  long.     For  the  left  end  of  the  horizontal  line  x  =  $", 
y  =  il". 
Prob.    2.  Given  a  horizontal  line  the  same  as  in  Prob.  i. 

(a)  At  the  left  end  of  this  line  erect  a  perpendicular  ij" 

long. 

(6)  At  2"  from  the  left  end  let  fall  a  perpendicular  i"  long. 
Prob.    3.  Given  a  horizontal  line  as  in  Prob.  i;    also  given  a  point 

*=3",y-al"- 

(a)  Let  fall  a  perpendicular  from  the  point  to  the  line, 
(ft)   Draw  a  line  i"  long  parallel  to  the  given  line  from  a 
point  di",  i"). 


Prob.  4.  Draw  a  line  making  an  angle  of  30°  with  the  horizontal,  3" 
long,  from  the  point  (i",  i");  divide  this  line  into  12  equal 
parts,  using  a  horizontal  line  as  an  auxiliary  line,  and  using 
the  two  triangles  to  draw  parallel  lines  (Art.  9). 

Prob.  5.  With  the  aid  of  the  two  triangles  lay  off  at  the  point  (J",  J"), 
on  a  horizontal  line,  angles  of  15°,  30°,  45°,  60°,  75°,  ane* 
90°,  making  the  radial  lines  terminate  in  the  sides  of  a 
2\"  square,  with  the  given  point  a  corner. 

Prob.    6.  (a)  Given  an  angle  of  60°  laid  off  from  a  horizontal  at  the 

point  (i",  J");  bisect  the  angle. 

(6)  Given  an  angle  of  90°  laid  off  from  a  horizontal  at  the 
point  (3",  3");  trisect  the  angle. 


PLATE  5. 


L 


v' 

X 


\ 


T 


,<"  I 


\\ 

\ 
\ 
\ 


V 


29 


30 


MECHANICAL   DRAWING. 


PLATE  6. 


Prob.  7.  With  a  radius  of  ij"  draw  an  arc  through  the  two  points 
(il",  i")  and  (J",  2j"). 

Prob.  8.  Draw  an  arc  through  the  three  given  points  (I",  2"),  (if", 
3"),  and  (3",  1"). 

Prob.  9.  With  a  radius  of  i"  and  center  at  (ii",  ij")  draw  a  circle. 
From  a  given  point  (3" ,  3")  draw  two  tangents  to  the 
circle. 

(Suggestion:  Bisect  the  line  joining  the  given  point  and  the 
center  of  the  circle,  and  with  this  point  of  bisection  as  a 
center  and  a  radius  equal  to  half  the  length  of  the  line 
describe  an  arc:  this  arc  will  cut  the  given  circle  in  the 
points  of  tangency.) 

Prob.  10.  With  a  radius  of  i"  and  center  at  (ii",  ii")  draw  a  circle, 
and  with  a  radius  of  J"  and  center  at  (z\",  2j")  draw 
another  circle.  Draw  two  tangents,  one  of  which  will  cut 
the  line  of  centers  between  the  circles  and  the  other  outside, 
if  produced.  Find  the  points  of  tangency  and  join  them. 
(Suggestion:  For  the  first  tangent  draw  a  circle  concentric 
with  the  larger  circle  and  with  a  radius  equal  to  the  differ- 
ence between  the  radii  of  the  two  given  circles;  next  draw 
a  tangent  to  this  auxiliary  circle  from  the  center  of  the 


smaller  circle  as  in  Prob.  9.  The  required  tangent  will  be 
parallel  to  this  tangent,  and  the  radii  of  contact  will  be 
perpendicular  to  it.  For  the  second  tangent  draw  a  circle 
concentric  as  before  with  a  radius  equal  to  the  sum  of  the 
radii  of  the  given  circles,  and  proceed  in  a  similar  manner 
as  in  the  other  case.) 

Prob.  ii.  With  center  at  (ij",  i}")  draw  a  circle  through  the  point 
(2",  2").  •  Draw  two  circles,  each  of  f"  radius,  tangent 
to  the  given  circle  at  the  given  point,  one  internal  and  the 
other  external. 

Prob.  12.  With  a  center  at  (W,  it")  draw  an  arc  A,  passing  through 
the  point  (if,  2\"),  and  with  a  center  at  (2}",  ij")  and 
radius  =  H"  draw  a  circle  B.  It  is  required  to  draw  a 
circle  tangent  to  the  given  arc  and  the  given  circle  at  the 
given  point  on  the  arc.  (Suggestion:  Draw  an  arc  concen- 
tric with  the  A  arc  having  a  radius  equal  to  the  difference 
between  the  radii  of  the  given  arc  and  circle  respectively; 
join  the  center  of  the  B  circle  and  the  point  where  the  radius 
of  the  A  arc  through  the  given  point  meets  the  auxiliary 
arc;  bisect  this  line.  The  center  of  the  required  circle 
•  will  be  at  the  intersection  of  these  two  lines.) 


GEOMETRIC    CONSTRUCTIONS,    AND    CONSTRUCTION   AND    USE    OF   SCALES. 


31 


PLATE  7. 


Prob.  13.  Given  three  points  (i",  i"),  (i}",   af"),  and  (3",   if). 

Draw  a  circumference  tangent  to  the  three  sides  of  the 

triangle  formed  by  joining  the  three  points. 
Prob.  14.  Given  a  horizontal  line  2\"  long  drawn  from  the  point  x  =  J", 

y  =  J"  to  the  right. 
Construct  on  this  line  an  equilateral  triangle  and  draw  three 

circles  tangent  to  the  sides  of  the  triangle  and  to  each  other. 
Prob.  15.  Inscribe  an  octagon  within  a  given  square  whose  side  is  2j"; 

the  lower  left-hand  corner  of  the  square  is  at  (J",  J")- 

(Suggestion:   With  each  corner  as  a  center  and  a  radius 

equal  to  half  the  diagonal  draw  arcs  cutting  the  sides  of 

the  square). 
Prob.  16.  Inscribe  a  regular  hexagon  in  a  circle  of  ij"  radius  whose 

center  is  at  (if",  i}").     (Suggestion:   The  radius  of  the 

circle  equals  the  side  of  the  hexagon.) 


Prob.  17.  Inscribe  a  regular  pentagon  in  a  circle  of  equal  radius  and 
like  position  as  that  in  Prob.  16.  Draw  two  diameters 
AB  and  DE,  perpendicular  to  each  other.  Call  C  the 
center  of  the  circle.  Bisect  AC  in  the  point  F;  with  F 
as  a  center  and  FD  as  a  radius  describe  an  arc  cutting 
AB  at  G,  then  GD  =the  length  of  the  side  of  the  required 
pentagon. 

Prob.  18.  Draw  a  regular  polygon  of  any  number  of  sides  (say  seven) 
on  a  given  base  AB  lA"  long  and  horizontal,  whose  middle 
point  is  (il",  A")-  With  AB  as  a  radius  and  A  as  a  center 
describe  a  semicircle  and  divide  it  (in  this  case)  into  seven 
equal  spaces;  join  A  and  the  second  point  of  division,  2, 
then  this  line  will  be  a  second  side  of  the  polygon.  By 
Prob.  8  find  the  center  of  the  circumscribing  circle  through 
the  points  A,  B,  and  a. 


32 


MECHANICAL   DRAWING. 


PLATE  8. 


Prob.  19  Connect  the  following  points  by  a  continuous  curve,  the 
several  parts  of  which  are  arcs  of  circles  tangent  to  each 
other,  the  first  arc  being  a  semicircle:  (fa",  ij");  *fa", 

i");  (aj",  B");  (ah",  ij'0;  (H",  2j");  (M",  art"); 

(2l"f  art");   (3",  »n;  and  (3fV',  if"). 

Prob.  20.  Draw  an  "egg  oval"  on  a  circumference  whose  diameter  is 
2"  and  whose  center  is  (ij",  if").  With  AB  as  a  radius 
and  center  at  A  describe  the  arc  BE,  and  with  DE  as  a 
radius  and  center  at  D  describe  the  arc  EE'. 

Prob.  21.  Draw  a  parabola  on  a  horizontal  axis  2}"  long  and  a  base  2" 
long;  the  intersection  of  the  base  and  axis  to  be  the  point 
(!"»  Il")-  Divide  AB  and  BD  each  into  the  same  number 
of  equal  spaces  and  number  the  points  of  division  as  shown. 
Draw  horizontal  lines  through  the  points  of  division  on 
AB,  and  join  A  with  the  points  of  division  on  BD;  where 
the  lines  correspondingly  numbered  intersect  are  points  on 
the  curve.  These  points  are  joined  first  by  an  irregular 
curve  drawn  free-hand  in  pencil  and  then  in  ink,  using 
the  scroll  as  a  guide  for  the  ruling-pen. 


Prob.  22.  Draw  an  ellipse  whose  major  axis  is  2\"  long  and  whose 
minor  axis  is  ij"  long  ;  the  intersection  of  the  two  axes  to 
be  the  point  x  =  if",  y  =  if"!  Divide  AB  and  AC  each 
into  the  same  number  of  equal  spaces,  join  D'  with  the 
points  of  division  on  AC,  and  D  with  the  points  of  division 
on  AB  ;  where  the  lines  correspondingly  numbered  meet 
are  points  on  the  curve. 

Prob.  23.  Draw  an  ellipse  on  axes  the  same  as  given  in  Prob.  22  by 
another  method  as  indicated  in  the  plate. 

Prob.  24.  Draw  an  hyperbola  on  a  horizontal  axis  i"  long  whose  mid- 
dle point  isx(if",  if").  Two  methods  of  construction 
are  shown,  one  on  the  left  by  a  method  similar  to  those 
used  in  finding  points  on  the  parabola  and  ellipse,  and 
the  other  on  the  right  as  follows:  Find  F  and  F'as  shown, 
take  any  distance,  as  Aa,  for  a  radius,  and  with  F'  as  a 
center  describe  an  arc  which  passes  through  G,  then  with 
the  difference  between  Aa  and  AA'  as  a  radius  and  F  as 
a  center  describe  an  arc  cutting  the  first  arc  in  G.  G  is  a 
point  of  the  curve. 


PLATE  8. 


19 


r 


21 


r 


~      B. 


J    L 


>A 


r 


22 


~l 


r 


J     L 


23 


r 


24 


Dl    2    38 


J-   L 


J 


33 


MECHANICAL   DRAWING. 


PLATE  9. 


31.  Construction  and  Uses  of  Scales.* 

1.  Construct  scales  of  J"  =  i',  J"  =  i'  and  ij"  =  i'.      Draw  three  heavy 

horizontal  lines  3"  long,  beginning  at  #  =  }",  and  i"  apart,  the 
first  line  to  beat  y  =  a\";  n  "  above  each  heavy  line  draw  a  parallel, 
light  line.  Mark  and  figure  the  scales  as  shown  in  the  accompany- 
ing plate. 

2.  Construct  a  diagonal  scale  of  TV,  or  |"  =  i  yd.,  to  show  yards,  feet, 

and  inches.  Draw  a  rectangle  3"  horizontal,  i"  vertical,  the 
left  side  at  x  =  \"  and  the  bottom  side  at  y  =  \".  Divide  the 
left  side  into  twelve  equal  spaces,  and  the  bottom  side  into  four 
equal  spaces  each  f  "  long,  and  the  first  of  these  spaces  to  the  left 
into  three  equal  spaces.  Mark  and  figure  as  in  the  accompanying 
plate. 

(a)  Lay  off  on  a  horizontal  line  drawn  from  (J"  2ft")  3  yds. 
i  ft.  5  in.  by  this  scale.  Place  one  point  of  the  spring  dividers 
at  the  intersection  of  the  vertical  line  through  3,  and  the  fifth 
horizontal  line  from  the  bottom,  and  the  other  point  at  the  inter- 
section of  the  same  horizontal  and  the  diagonal  through  i. 

(6)  Lay  off  on  another  horizontal  line  drawn  from  x  =  i",  y  =  ?l", 
i  yd.  2  ft.  10  in.  All  lines  called  "dimension-lines"  which  are 
drawn  to  show  the  limits  of  a  measurement  should  be  drawn 
light,  full  in  red  ink  (shown  in  the  plate  as  broken  lines),  the  arrow- 
points  at  the  extremities,  and  the  numerals  in  black  ink. 

3.  Construct  a  diagonal  scale  of  inches,  tenths,  and  hundredths  of  an 

inch.  Draw  a  rectangle  as  in  2,  mark  and  figure  as  in  the  accom- 
panying plate. 

*  The  original  drawings  have  been  reduced  in  dimensions  in  reproducing 
Plates  9  and  10,  so  that  the  scales  and  distances  in  the  plates  are  propor- 
tional only  to  the  dimensions  noted  in  the  text. 


(a)  Lay  off  on  horizontal  lines,  situated  as  in  2,  the  distances  2.54" 

and 
(6)  1.46"  by  this  scale.     The  principle  is  the  same  as  in  2. 

4.  Construct  a  scale  for  reducing  to  f  and  I.     Draw  a  horizontal  line 

3"  long  from  the  point  x  —  \",  y  =  $"',  erect  a  perpendicular  at 
the  left  end  2$"  long  and  divide  the  perpendicular  into  five 
equal  parts;  join  the  third  and  fifth  divisions  with  the  right  end 
of  the  horizontal  line.  Mark  and  figure  as  in  the  accompanying 
plate. 

5.  Construct  a  square  whose  side  is  2j"  and  whose  lower  left-hand 

corner  is  at  x=l",  y=l".  By  the  reduction  scale  of  4  draw 
two  other  squares  inside  the  first,  symmetrically  placed,  one 
whose  sides  are  f  and  the  other  whose  sides  are  f  the  sides 
of  the  first  square.  AB  on  the  scale  =  2 J",  AC=|  AB,  and 
CB=|  AB. 

6.  Construct  a  Scale  of  Chords.     Draw  a  horizontal  line  3"  long  from 

the  point  x  =  \",  y=\".  At  a  point  #  =  1.12"  (use  the  diagonal 
scale  of  2)  on  this  line  as  a  center  and  with  a  radius  =  2. 13"  draw 
a  quarter  of  a  circumference.  Divide  this  arc  into  9  equal  parts 
and  transfer  the  chords  of  these  arcs  to  the  horizontal  line.  All 
lines  except  those  of  the  scale  proper  are  drawn  in  red  ink. 
Mark  and  figure  as  in  the  accompanying  plate.  (Suggestion:  To 
measure  a  given  angle  with  this  scale,  using  a  radius  equal  to  the 
chord  of  60°  on  the  scale,  describe  an  arc  from  the  vertex  of  the 
angle  as  a  center,  measure  the  length  of  the  chord  on  this  arc 
between  the  sides  of  the  angle  and  apply  it  to  the  scale.  To  lay 
off  an  angle  from  a  given  line  at  a  point  on  the  line,  describe  an 
arc  from  the  point  as  a  center  and  a  radius  equal  to  the  chord  of 
60°  on  the  scale,  and  on  this  arc  lay  off  the  chord  of  the  required 
number  of  degrees  as  obtained  from  the  scale. 


PLATE  9. 


2 


.3 


Scale  iW 


9630  1  2  3  4  5 


|» 1  yd.  210"— 

3yd.  l's" 


Scale-|"=r 


t    t     3   0 


Scale  \£-  1' 


10     8       0      4       2      0 


Scale  -453-,  or- j-  =  l  yd. 


3210 


1    " 
Diagonal  Scale  to  755 


o   i   a   o 


Scale  o<  Chords  , 


Reduction  Scale 
I  and  f 


90    80    70     60      50      40      30       80      10        0 


35 


36 


MECHANICAL   DRAWING. 


PLATE  10. 


7.  Scale  y=i'  (use  the  flat  scale).     With  the  longer  side  hori- 

zontal draw  a  rectangle  7'  by  10'. 
For  the  lower  left  corner  x=2r,  y=z'  6". 

8.  Scale  i"=  i'.     From  the  point  (4',  8')  draw  a  horizontal  line 

20'  long;  upon  this  as  a  base  construct  a  triangle  whose 
other  sides  are  14'  and  18'.  Measure  the  angles  with  the 
protractor  and  substitute  the  number  of  degrees  for  the 
question-mark. 

9.  Scale  3^5  or  i"=3o'  (use  the  triangular  scale  of  30  parts). 

Draw  a  polygon;  ist,  from  the  point  (9',  20')  draw  a 
horizontal  line  87'  long;  2d,  from  the  same  point  a  line 
72';  jd  (continuing  around  to  the  right),  a  line  42';  the 
fourth  side  to  close  the  polygon.  The  angle  between  the 
first  and  second  sides  =63°;  between  the  second  and  third 
sides =90°. 

Determine  with  the  scale  the  fourth  side,  and  with  the  pro- 
tractor the  two  adjacent  angles,  and  insert  them  in  their 
proper  places. 

10.  Scale  y=i'  (use  the  diagonal  scale  in  Plate  9).  From  the 
point  #=0.62',  y=2r  draw  a  horizontal  line  5.96'  long; 
upon  this  as  a  base  construct  a  triangle  whose  other  sides 
are  4.08'  and  3.64'  respectively. 


Measure  the  angles  with  the  scale  of  chords  in  Plate  9. 

11.  Scalei"=i'.     Construct  a  polygon. 
For  the  point  A,  x=i'  10",  y=T,'  4". 

Take  AB  on  a  horizontal  line  to  the  right,  10'  3$". 

Take  AC  =  9'  9"  on  a  line  making  an  angle  of  30°  with  AB 

(use  the  scale  of  chords). 
Take  AD =8'  9"  on  a  line  making  an  angle  of  45°  with 

AC. 
Measure  the  sides  DC  and  CB  with  the  scale  and  the  angles 

ADC,  DCB,  and  ABC  with  the  scale  of  chords. 

12.  Scale  |  (use  a  reduction-scale  made  on  a  separate  piece  of 

paper).     Construct  a  polygon. 
For  the  point  A,  x=^",  y=z"  • 
Take  AB  on  a  horizontal  line  to  the  right,  6". 
Take  AC=io£"  on  a  line  making  an  angle  .of  26°  with  AB 

(use  the  protractor). 

Take  AD=  io|"  on  a  line  making  an  angle  of  38°  with  AC. 
Take  AE=8Jf"  on  a  line  making  an  angle  of  41°  with  AD. 
Take  AF=4"  on  a  line  making  an  angle  of  30°  with  AE. 
Measure  the  sides  BC,  CD,  DE,  and  EF  with  the  same  scale; 

also  the  angles  ABC,  BCD,  CDE,  DEF,  and  EFA  with 

the  protractor. 


PLATE   10. 


•10- 


.  Scale:  £=!' 

10 


S J.96'. ^ 


8 


h- 35 


-20'- 


11 


Af, 


Scale:  £— 1' 


12 


Scale: 


37 


CHAPTER  IV. 
PROJECTIONS. 


32.  Definitions. — In  representing  an  object  by  draw- 
ings on  a  plane  surface,  if  the  projecting  lines  of  the  several 
points  are  perpendicular  to  the  plane  of  projection  the 
projection  is  called  orthographic,  and  such  drawings  are 
used  as  working  drawings.  This  chapter  treats  of  ortho- 
graphic projections  only. 

The  projection  of  an  object  may  be  made  on  one  plane 
only,  or  on  several  planes  in  order  to  show  its  true 
form;  generally  at  least  two  projections  are  made,  one 
on  a  horizontal  and  the  other  on  a  vertical  plane.  These 
planes  are  called  the  principal  planes  of  projection.  A 
plane  of  projection  which  is  perpendicular  to  both  the 
horizontal  and  vertical  planes  is  called  a  perpendicular 
or  profile  plane,  and  a  plane  of  projection  having  any 
other  position  with  reference  to  the  horizontal  and  ver- 
tical planes  is  known  as  a  supplementary  plane. 

It  is  convenient  to  consider  that  the  object  may  be 
placed  in  any  desired  position  with  relation  to  the  hori- 
zontal plane  of  projection;  the  vertical  plane  of  projec- 
tion may  then  be  taken  perpendicular  to  the  horizontal 
plane  in  any  possible  relation  to  the  object.  The  profile 


plane  is  determined  in  its  position  as  perpendicular  to 
both  the  horizontal  and  vertical  planes. 

The  plane  of  the  paper  is  assumed  to  coincide  with 
each  plane  of  projection  in  turn,  as  the  drawing  is  made 
on  that  particular  plane. 

33.  Conventions. — In  these  drawings  each  object  to 
be  represented  by  its  projections  will  be  assumed  in  the 
third  angle,  i.e.,  below  the  horizontal  and  behind  the 
vertical  plane.  The  horizontal  projection  (top  view  br 
plan)  will  be  shown  above  the  vertical  projection  (froiit 
view  or  elevation). 

H  will  be  used  to  designate  the  horizontal  plane  of 
projection,  V  the  vertical  plane  of  projection,  and  I5  the 
profile,  or  perpendicular,  plane  of  projection.  When  i  a 
supplemental  plane  of  projection  is  used,  it  will  be  referred 
to  as  S. 

The  lower  and  left  border-lines  of  the  plate  are  respect- 
ively reference-axes  for  locating  points  on  the  drawing 
(Art.  19). 

Any  point  referred  to  on  the  object  will  be  designated 

by  a  capital  letter,  its  projection  on  the  several  planes  by 

38 


PROJECTIONS. 


39 


the  same  capital  letter  with  a  subscript  letter  designating 
which  plane,  thus:  A^  represents  its  projection  on  H; 
Av,  its  projection  on  V;  Af  and  AJ(  its  projections  on 
P  and  S  respectively.  For  the  same  point  in  a  develop- 
ment (pattern)  the  same  capital  letter  will  be  used  with- 
out a  subscript. 

Whenever  a  line  in  a  drawing  represents  an  edge  of 
a  face  which  is  in  view  (the  planes  of  projection  are  con- 
sidered transparent),  it  is  to  be  drawn  full  in  black. 
If  it  is  not  in  view,  it  is  invariably  to  be  made  a  light 
dash-line  in  black.  All  construction-lines  when  shown 
in  the  finished  drawing  must  be  light,  full  lines  in  red. 
Lines  connecting  the  different  projections  of  the  same 
point  must  be  light  dot-lines  in  red.  All  dimension-lines 
should  be  drawn  as  stated  in  Art.  31,  2  (b). 

34.  Shade-lines. — When  the  object  to  be  represented 
is  made  up  of  plane  surfaces  or  faces,  a  distinction  is  made 
in  the  intensity  of  the  lines  representing  edges  that  are  in 
view.  Whenever  a  line  represents  an  edge  joining  two 
faces  both  in  the  light  or  both  in  the  shade,  it  is  drawn  light. 
Whenever  it  joins  two  faces,  one  in  the  light  and  the  other 
in  the  shade,  it  is  made  heavier.  These  heavier  lines  are 
called  shade-lines  (sometimes  called  shadow-lines). 

To  determine  accurately  when  a  face  is  in  the  shade  or 
in  the  light  involves  principles  of  Descriptive  Geometry 
with  which,  for  this  course,  it  is  not  assumed  the  student 
is  familiar.  However,  an  example  or  two  may  make 
plain  the  method  of  determining  shade-lines.  It  is  con- 


ventional to  regard  the  rays  of  light  as  parallel,  and 
coming  over  the  left  shoulder  in  the  direction  which  the 
diagonal  of  a  cube  has  when  the  faces  of  the  cube  are 
parallel  to  the  H  and  V  planes.  The  projections  of  rays 
on  H  would  then  have  a  direction  making  an  angle  of  45 
degrees,  with  a  horizontal  line  inclining  upwards  to  the 
right,  and  the  projections  on  V  an  angle  of  45  degrees 
inclining  downwards  to  the  right. 

In  some  drawings  it  is  easy  to  determine  by  inspection 
which  faces  are  in  the  light  and  which  are  in  the  shade. 
For  example,  in  Fig.  7  the  hexagonal  prism  has  its  bases 
parallel  to  H,  and  V  is  taken  parallel  to  a  face.  R/,  and 
R0  show  the  direction  of  the  projections  of  the  rays  of 
light  on  H  and  V  respectively. 

Draw  the  projection  on  H  of  a  ray  of  light,  as  Qh, 
moving  in  the  direction  of  the  arrow;  then  any  line,  as 
Q0,  drawn  parallel  to  Re  may  be  taken  as  the  projection 
of  the  same  ray  on  V.  This  ray  of  light,  Q,  pierces  the 
face  of  the  prism,  whose  projection  on  H  is  the  line 
Afc  Bfc,  and  its  projection  on  V  the  rectangle  Av  B0  Av'  B0', 
in  the  point  whose  projection  on  H  is  Gfc,  and  whose 
projection  on  V  is  Gv.  If  produced  it  would  afterwards 
pierce  the  face  BA  0,-B,,  C0  B/  CB'  in  the  point  Kh-Kv, 
showing  that  this  face  is  in  the  shade  while  the  first- 
mentioned  face  is  in  the  light. 

Again,  the  ray  Sft-S0  pierces  the  back  face  Fh  E^-F,,  A0 
F/  A/  before  piercing  any  other  face,  showing  that  the 
face  is  in  the  light. 


40 


MECHANICAL   DRAWING. 


In  a  similar  manner  it  may  be  determined  which  of 
the  other  faces  are  in  the  light  and  which  are  in  the 
shade. 

When  the  same  prism  is  tipped  as  in  Fig.  8  to  find 
whether  the  face  A  B  E  F  is  in  the  light  or  in  the  shade 
by  the  intervention  of  the  face  A'  B'  E  F  between  it  and 
the  source  of  light,  proceed  as  follows:  Draw  the  projec- 
tion on  H  of  a  ray  of  light,  as  the  line  RA,  making  an 
angle  of  45  degrees  with  a  horizontal  line,  and  crossing 
the  H  projections  of  both  faces.  From  the  point  H/,, 
where  this  line  crosses  the  line  AA  B&,  let  fall  a  perpen- 
dicular meeting  A/  B/  at  H,,;  similarly  find  Mv  and 
N,,.  Join  NB  with  Hv  and  M,,  respectively.  The  line  N  H 
lies  in  the  face  A  B  E  F,  and  M  N  in  the  face  A'  B'  E  F. 
Now,  if  from  any  point  on  MB  Ne,  as  G0,  a  line  Rv  be 
drawn  making  an  angle  of  45  degrees  with  a  horizontal 
line,  inclining  downwards  to  the  right,  it  will  be  the 
projection  on  V  of  a  ray  of  light,  which  is  one  of  the 
many  rays  of  which  RA  is  the  common  projection  on  H. 
If  R,,,  when  prolonged  in  the  direction  of  the  arrow, 
meets  the  line  Nv  H,,,  prolonged  if  necessary,  it  shows 
that  this  ray  of  light  pierces  the  face  A'  B'  E  F  before 
it  does  the  face  A  B  E  F;  which  means,  therefore, 
that  the  face  A  B  E  F  is  in  the  shade.  If,  however,  R0 
when  drawn  in  the  opposite  direction  had  met  the  line 


N0  H,,,  it  would  have  shown  the  face  A  B  E  F  to  be  in 
the  light. 

Again,  to  find  whether  the  face  A'  B'  C'  D'  prevents 
the  light  from  shining  on  the  face  D'  C'  K  L,  use  the 
same  horizontal  projection  of  a  ray  of  light,  RA.  Where 
Rh  meets  the  line  A.k'  Bh'  at  MA,  let  fall  a  perpendicular 
meeting  A0'  BJ0'  at  Mv;  and  where  it  meets  the  line 
Ch'  Dh'  at  MA  let  fall  a  perpendicular  meeting  the  line 
C/  D0'  at  Pv;  also  where  it  meets  KA  Lfc  at  Nfc  let  fall 
a  perpendicular  meeting  Ku  Lv  at  Q0.  Join  P,,  with 
Mc  and  Qv  respectively.  At  any  point  on  Mr  P0,  as 
Ww>  draw  R/  parallel  to  R0.  It  does  not  intersect 
P0  Q,,;  therefore  the  face  D'  C'  K  L  is  in  the  light. 

By  inspection  it  is  clear  that  the  end  face,  or  base,  to 
the  left,  and  also  the  face  A'  B'  C'  D',  are  in  the  light. 
Also  it  is  clear  that  the  other  base  and  the  face  A  B  C  D 
are  in  the  shade. 

In  the  examples  given  it  was  shown  that  the  face 
A  B  E  F  is  in  the  shade,  and  that  the  face  D'  C'  K  L 
is  in  the  light.  Similarly  find  that  the  remaining  face 
C  D  L  K  is  in  the  shade  by  the  intervention  of  the  face 
C'  D'  L  K. 

In  the  drawings  to  follow,  the  shade-lines  as  deter- 
mined in  the  projections  on  H  and  V  will  not  change 
in  the  projections  on  the  P  and  S  planes. 


PROJECTIONS. 


41 


F;      A; 


B'  Ci 


FIG.  7. 


FIG.  8 


42 


MECHANICAL   DRAWING. 


PLATE  11. 


35.  The  Cube. — Draw  the  projections  on  H  and  V  of 
a  cube  whose  edge  is  2",  in  each  of  the  following  three 
positions: 

(a)  When  one  face  of  the  cube  is  parallel  to  H,  and  V 

is  parallel  to  an  adjacent  face. 
In  the  projection  on  H,  the  nearer,  left  corner  of 

the  top  face,  Ah,  is  at  (i",  5"). 
In  the  projection  on  V,  Av  is  at  (i",  3.5"). 

(b)  When  one  face  of  the  cube  is  parallel  to  H,  and  V 

makes  an  angle  of  30  degrees  with  an  adjacent 

face. 
In  the  projection  on  H,  the  nearest  corner,  AA,  is 

at  (5.25",  4.25"). 
In  the  projection  on  V,  Av  is  at  (5.25",  3.5")- 

(c)  When  the  top  edge,  A  K,  of  the  cube  is  parallel  to 

H,  the  adjacent  faces  each  making  an  angle  of 
45  degrees  with  H;  and  V  makes  an  angle  of  60 
degrees  with  the  edge  A  K.  Draw  the  projection 
on  H  first. 


In  the  projection  on  H,  Ak  is  at  (9.5",  5.5")- 
In  the  projection  on  V,  A,  is  at  (9.5",  4.25"). 
The  front  face  is  perpendicular  to  H,  therefore  its 
projection  on  H  is  a  straight  line  Ah'  Bh',  equal 
to  the  diagonal  of  a  face.     In  the  projection  of 
this   face  on  V  the  distance  Ar  B0'  is  equal  to 
the  diagonal  of  a  face. 

The  edge  A  K  is  parallel  to  H,  therefore  its  projec- 
tion on  V  is  the  horizontal  line  Av  Kv,  and  its 
length  is  equal   to   the   perpendicular   distance 
between  the  projecting  lines  A,,  Ah  and  Kj,  Kfc. 
All  edges  that  are  parallel  to  each  other  are  pro- 
jected in  lines  that  are  parallel  on  both  H  and  V. 
36.  Sections. — If  an  object  or  body  be  intersected  or 
cut  by  a  plane,  and  the  part  of  the  body  on  either  side 
of  the  cutting  plane  be  removed,  the  surface  thus  exposed 
-  is  called  a  section.    The  plane  is  called  a  section  plane, 
and  its  intersection  with  a  plane  of  projection  is  called 
its  trace  on  that  plane  and  is  indicated  by  a  black  dash- 
and-two-dot  line. 


PLATE  11". 


CUBE 


Ad- 


43 


44 


MECHANICAL  DRAWING. 


PLATE  12. 


In  this  plate  copy  in  pencil  in  the  same  positions  the 
projections  of  the  cube  as  they  are  on  the  preceding 
plate,  and  show  sections  as  follows: 

(a)  In  this  position  pass  a  section  plane  perpendicu- 

lar to  V  and  making  with  H  an  angle  of  60 

degrees. 
In  the  projection  on  V,  cut  the  edge  Av  B0,  f "  to 

the  right  of  A0,  and  show  the  section  projected 

onH. 
The  wedge  projected  on  V  in  Av  Dv  Cv  is  removed 

and  the  section  is  shown  on  H  in  Ch  Gh  Dh  Kh. 

(b)  In  this  position  pass  a  section  plane  perpendicular 

to  H  and  parallel  to  V. 

In  the  projection  on  H,  let  Eh  Fh  be  4.75"  above 
the  horizontal  axis  of  the  plate.  Show  the  sec- 
tion projected  on  V.  The  wedge  projected  on 
H  as  Ah  Eh  Fh  is  removed  and  the  section  is 
shown  on  V  in  F.v  Lv  Fv  Ns. 

(c)  In  this  position  pass  a  section  plane  perpendicular 

to  H  and  parallel  to  V. 


In  the  projection  on  H,  let  Q,  Dh  be  5.75"  above 
the  horizontal  axis  of  the  plate.  Show  the  sec- 
tion projected  on  V. 

In  the  projection  on  H.all  that  portion  of  the  cube 
in  front  of  the  plane  Ch  Dh  is  removed.  The 
edge  whose  projection  on  H  is  A/,  BA  is  cut  at  C, 
which  is  shown  at  CB  in  the  projection  on  V. 
The  front  face  o*  the  cube  is  cut  in  a  vertical 
line;  its  projection  qn  V  is  C0  Ge.  The  edge 
whose  projection  on  H  is  Ah  Fh  is  cut  at  E, 
shown  as  E0  in  the  projection  on  V.  Similarly 
DB  is  found.  The  boundary  of  the  section  is 
obtained  by  joining  consecutively  the  points  so 
determined. 

Notes:  In  the  projection  on  which  the  section  is 
shown,  ink  in  no  lines  representing  parts  re- 
moved. "  Section-lines"  are  parallel  lines  drawn 
on  a  section,  about  &"  apart,  making  some 
angle  with  a  horizontal  line,  usually  45  degrees. 
The  trace  of  a  section  plane  should  be  a  dash- 
and-two-dot  line  drawn,  in  black. 


PLATE  12. 


SECTIONS 


G* 


C,, 


A*  DA 


A., 


45 


46 


MECHANICAL    DRAWING. 


PLATE  13. 


Sections  (continued). — Draw  the  projections  on  H  and 
V  of  a  cube  whose  edge  is  2",  and  which  has  a  square 
hole  cut  through  it  between  two  opposite  faces.  The 
side  of  the  square  hole  is  i". 

(a)  One  face  of  the  cube  is  parallel  to  H,  and  V  makes 

with  an  adjacent  face  an  angle  of  15  degrees. 

The  edges  of  the  hole  running  through  the  cube 

are  parallel  to  V. 
In  the  projection  on  H,  the  nearer,  left  corner  of  the 

top  face,  Ah,  is  at  (1.5",  5"). 
In  the  projection  on  V,  Av  is  at  (1.5",  3-5")- 
Draw  the  projection  on  H  first. 

(b)  Two    adjacent   faces  each  make  an   angle  of  45 

degrees  with  H,  and  V  is  parallel  to  a  face.     The 

edges  of  the  hole  which  run  through  the  cube  are 

perpendicular  to  V. 
In  the  projection  on  H,  the  highest,  front  corner, 

A,,  is  at  (6.5",  5")- 

In  the  projection  on  V,  Av  (not  shown  but  coincid- 
ing  with  E.')  is  at  (6.5",  4.3"). 


Cut  this  position  of  the  cube  by  a  section  plane 
perpendicular  to  H  and  making  an  angle  with 
V  of  30  degrees,  inclined  backwards  to  the  right, 
and  cutting  the  front  face  one-half  inch  to  the 
right  of  the  corner  B. 

Show  the  section  on  the  projection  on  V. 

Draw  the  projection  on  V  first. 

(c)  Draw  a  projection  on  the  profile  plane  to  the  right 
of  this  second  position  of  the  cube,  and  show 
the  section. 

In  the  projection  on  P,  AP  is  at  (9",  4.3"). 

H,,  E/  =  Hfc  EA;  Gf  Kf  =  Gh  Kfc;  Ff  Kf  =  Fh  LA; 
D,K,  =  DfcMfc;  E/E,-E.'E.;  C,  B,  =  theper- 
pendicular  distance  between  the  projecting  lines 
of  C  and  B  on  P,  shown  by  the  perpendicular 


distance  between  C.. 


Cf  and  Bt 


Bp.      Similarly 


the  lengths  of  other  lines  in  the  projection  on  P 
may  be  found. 


PLATE  13. 


CUBE  WITH  HOLE 


I  I 


47 


48 


MECHANICAL   DRAWING. 


PLATE  14. 


37.  Rectangular  Prism. — Draw  the  projections  on  H 
and  V  of  a  rectangular  prism  whose  base  is  a  rectangle 
2"  by  1.5"  and  whose  height  is  3". 

(a)  When  the  base  is  parallel  to  H,  and  V  is  parallel  to 

the  faces  which  are  2"  wide. 
In  the  projection  on  H,  the  front,  left  corner  of  the 

upper  base,  Ah,  is  at  (i",  5.5")- 
In  the  projection  on  V,  Av  is  at  (i",  4.5")- 

(b)  When  the  narrowest  faces  make  with  H  an  angle 

of  30  degrees  (inclining  upwards  to  the  right), 
and  V  is  parallel  to  the  broadest  faces.  Draw 
the  projection  on  V  first. 

In  the  projection  on  H  the  lowest,  front  corner, 
Cto  is  at  (57",  5-5"). 


In  the  projection  on  V,  CB  is  at  (5.7",  1.5"). 

Cut  this  position  of  the  prism  by  a  section  plane 
perpendicular  to  H  and  making  an  angle  with 
V  of  30  degrees,  extending  backwards  to  the 
right.     Fh  is  \"  from  Bh. 
Show  the  section  in  the  projection  on  V. 
(c)  Draw  the  projection  of  this  second  position  on  the 
P  plane  to  the  right,  and  show  the  section. 

In  the  projection  on  P,  C^  is  at  (10",  1.5"). 

APEp  =  AhEh;  BfFf  =  BhFh;  D,Gp  =  the  per- 
pendicular distance  between  the  projecting  lines 
on  P  of  D  and  G,  shown  by  the  perpendicular 
distance  between  DB  Df  and  Ge  Gf. 


PLATE  14. 


RECTANGULAR  PRISM 


IB, 


...-"  F, 


49 


50 


MECHANICAL    DRAWING. 


PLATE  15. 


38.  Supplemental  Projections. — Draw  the  projections  on 
H  and  V  of  the  same  prism  that  is  given  in  Plate  14. 

(a)  When  the  base  is  parallel  to  H,  and  V  makes  an 

angle  of  30  degrees  with  the  widest  face. 

In  the  projection  on  H,  the  nearest  corner,  A*,  of 
the  upper  base  is  at  (7",  5").  Draw  the  pro- 
jection on  H  first. 

In  the  projection  on  V,  Ao  is  at  (7",  4.5")- 

(b)  Draw  to  the  right  a  profile  of  this  prism. 
In  the  projection  on  P,  A.p  is  at  (9",  4.5")- 
Assume  in  (a)  a  section  plane  perpendicular  to  V, 

and  making  an  angle  of  30  degrees  with  H.  Let 
the  intersection  of  this  plane  with  V  (called  the 
V  trace)  cut  the  edge  farthest  to  the  right  at 
i.i"  down  from  the  top.  Show  the  section  only 
on  the  P  projection  in  (b). 

(c)  Draw  a  supplemental  projection  of  the   prism  to 


the  left  on  a  plane  perpendicular  to  V,  and 
making  an  angle  of  60  degrees  with  H. 
In  the  projection  on  S,  As  is  at  (4.5",  5.9")- 
In  the  projection  on  S,  the  projected  length  of  the 
edge  A  K  on  S  is  found  by  projecting  the  line 
Av  Kv  on  a  line  through  A.s,  making  an  angle  of 
60  degrees  with  a  horizontal  line.  The  point  a 
is  the  intersection  of  a  horizontal  through  AA 
and  the  60°  line  through  As.  The  projections  of 
the  other  parallel  edges  are  equal  and  parallel 
to  As  Ks.  The  perpendicular  distance,  a  b,  be- 
tween the  edges  projected  in  As  K{  and  Bs  Ls  is 
found  by  projecting  the  line  A/,  Bh  on  a  line 
perpendicular  to  a  horizontal,  as  a  b.  Simi- 
larly the  perpendicular  between  the  lines  B^  Ls 
andCj  Nj  =  c  6  =  Cx  &!.  The  projections  of  oppo- 
site sides  of  the  prism  are  equal  parallelograms. 


SUPPLEMENTARY  PROJECTION 


PLATE  15. 


n  ">  — 

HX 

^ 

^N 

.^ 

£>.... 

51 


52 


MECHANICAL   DRAWING. 


PLATE  16. 


39.  Hexagonal  Prism. — Draw  the  projections  on  H  and 
V  of  a  prism  3"  long,  whose  base  is  a  regular  hexagon 
with  a  side  of  i". 

(a)  When  the  base  is  parallel  to  H,  and  V  is  parallel  to 

a  face. 

In  the  projection  on  H,  the  upper  corner  of  the 
front  face,  Ah,  is  at  (3",  5").  Draw  the  projec- 
tion on  H  first. 

In  the  projection  on  V,  Ae  is  at  (3",  4.5")- 

(b)  Draw  a  projection  of  this  position  of  the  prism  on 

the  P  plane  to  the  right. 
In  the  projection  on  P,  Af  is  at  (4.25",  4.5"). 
The  perpendicular  distance  between  the  edges  of 

the  faces  in  front  of  the  P  plane,  as  Ap  B,,  =ab. 


(c)  Draw  the  projections  on  H  and  V  of  the  same 

prism  when   the   edges  make  an  angle  of   30 
degrees    with    H,    and  V    is    parallel    to    one 
face. 
In  the  projection  on  H  the  corner  A.h  is  at  (7.75", 

5"). 

In  the  projection  on  V,  Av  is  at  (7.75",  4")- 
Draw  the  projection  on  V  first.     (It  is  the  same 

figure  as  the  projection  on  V  in  (a).) 

(d)  Draw  the  projection  of  one-half  of  the  lower  base 

on  a  supplemental  plane  parallel  to  and  \" 
from  it,  as  Ds  Fs.  This  supplementary  projec- 
tion may  be  used  to  get  the  projection  on  H, 


HEXAGONAL  PRISM 


PLATE  16. 


53 


54 


MECHANICAL   DRAWING. 


PLATE  17. 


40.  Development. — Draw  again  the  projections  (c)  and 
(d)  of  the  previous  plate,  cut  the  prism  by  a  section 
plane,  and  develop  a  portion  of  the  prism. 

(a)  The  V  trace,  AB  DB,  of  the  section  plane  is  parallel 

to  H,  being  3.5"  from  the  horizontal  axis  of  the 
plate.  Show  the  section  on  the  horizontal  pro- 
jection. 

In  the  projection  on  H,  ~Fk  is  at  (2.25",  5"). 

In  the  projection  on  V,  Fv  is  at  (2.25",  4"). 

(b)  Develop,   or  make  a  pattern  of,   the  side   faces 

of  the  larger  part  of  this  prism,  that  part  be- 
low the  section  (omitting  the  base  and  the 
section). 

Suppose  the  prism  to  be  hollow.  Since  all  of  its 
faces  are  plane  surfaces,  they  may  be  made  to 
coincide  with  a  common  plane  by  causing  each 


face  in  turn  to  come  into  that  plane  by  rotation 
on  the  uniting  edges  in  succession. 

Let  K  be  at  (n",  4.25").  The  sides  of  the  base 
will  fall  in  the  line  H  H.  The  edges  of  the  side 
faces  will  be  perpendicular  to  the  base,  H  H,  and 
be  separated  by  a  distance  equal  to  the  side  of  the 
hexagon  of  the  base.  The  true  lengths  of  the 
edges  will  be  found  in  their  projections  on  V, 
because  they  are  parallel  to  V.  That  is,  D  H  = 
D,,  Hv,  etc.,  and  A'  L= Ar  L,,  for  the  same  reason. 

If  this  pattern  were  cut  out  of  the  paper  following 
its  outline  and  bent  on  the  lines  of  the  uniting 
edges,  it  would  take  the  form  of  the  lower  part 
of  the  prism.  It  could  thus  be  used  as  a  pat- 
tern to  mark  out  on  sheet  metal  or  other  thin 
material  the  surface  of  the  prism. 


DEVELOPMENT 


PLATE  17. 


55 


56 


MECHANICAL  DRAWING. 


PLATE  18. 


41.  Intersections. — In  the  intersection  of  surfaces  it  is 
required  to  show  the  projection  of  lines  which  are  com- 
mon to  both  surfaces,  that  is,  their  intersection. 

Draw  the  projections  of  two  prisms  intersecting  each 
other,  one  with  a  hexagonal  base  and  the  other  with  a 
square  base,  with  dimensions  as  shown  on  the  plate. 
(a)  The  base  of  the  hexagonal  prism  is  parallel  to  H, 
and  V  is  parallel  to  a  face.     Draw  the  projection 
on  H  first. 
In  the  projection  on  H,  the  center  of  the  top  base 

is  at  (2",  6"). 
In  the  projection  on  V,  the  center  of  the  top  base 

is  at  (2",  4"). 

The  other  prism  is  behind  the  first;   its  axis  inter- 
sects and  is  perpendicular  to  the  first  prism  and 
is  perpendicular  to  V. 
The  faces  make  an  angle  of  45  degrees  each  with  H. 


Draw  the  projection  on  V  first. 

In  the  projection  on  H,  the  middle  of  the  base  is  at 

(2",  7-75"). 
In  the  projection  on  V,  the  middle  of  the  base  is  at 

(2",  2.75"). 

(b)  Draw  a  projection  on  a  P  plane  to  the  right. 

In  the  projection  on  P,  the  middle  of  the  upper 
base  is  at  (5",  4"). 

Cpc"=Q,c  or  CpL#=QLA.  The  distance  from 
Bp  to  the  axial  line  Cfc"  is  equal  to  the  perpen- 
dicular distance  between  the  projecting  lines 
from  Bv  and  CB  on  Df  Ef. 

(c)  Develop  the  side  faces  of  the  hexagonal  prism  and 

trace  on  the  pattern  the  intersecting  lines  of  the 

two  prisms. 

The  corner  B'  of  the  upper  base  is  at  (11.5",  4.25"). 
B'B  =  B,,'B-   B'G-B.'G.:   B'C'=BfcCfc. 


PLATE  18. 


_ - 


I— 4s- 


INTERSECTION 


B;  c; 


f 

i 

*?— 

i 

/ 
/ 

\ 

/ 

\ 

/ 

\ 

t/ 

\ 

Br 

,<X 

„ 

- 

:<^c_0_ 

•>;  \ 

"as 

\       S, 

\ 

^v 

y 

GT 

V 

/ 

V 

iHt 


t" 

e' 


57 


58 


MECHANICAL   DRAWING. 


PLATE  19. 


42.  Oblique  Intersections.  —  Draw  the  projections  of 
two  prisms  intersecting  each  other  obliquely.  The  base 
of  each  is  a  regular  hexagon.  The  dimensions  are  shown 
in  the  plate. 

(a)  In  the  upright  prism  the  base  is  parallel  to  H,  and 

V  is  parallel  to  a  face. 
In  the  projection  on  H,  the  center  of  the  upper 

base  is  at  (6",  6"). 
In  the  projection  on  V,  the  center  of  the  upper 

base  is  at  (6",  4.25"). 
Draw  the   projection   on  H  first  of  the  upright 

prism. 

The  axis  of  the  oblique  prism  inclines  to  H  at  an 
angle  of  45  degrees  downwards  to  the  right  and 
is  parallel  to  V.  The  plane  of  its  base  is  perpen- 
dicular to  its  edges. 

In  the  projection  on  H,  the  middle  of  the  base,  Q,, 
is  at  (4t",  6"). 


Draw  the  projection  of  one-half  of  the  base  of  the 
oblique  prism  on  a  supplemental  plane,  parallel 
to  the  base  and  ±"  from  it. 

In  the  projection  on  S,  the  center  of  the  base,  C, 
is  at  (4",  3-75"). 

Draw  the  projection  on  the  S  plane  first,  on  the  H 
plane  second,  and  finally  on  the  V  plane. 

(b)  Draw  the  projections  of  these  prisms  on  a  P  plane 

to  the  left.  The  axial  line,  a  b,  is  2"  from  the 
vertical  axis  of  the  plate.  Study  the  figures  on 
the  plate  to  determine  the  projected  distances 
between  edges  and  their  ends. 

(c)  Develop  the  oblique  prism  and  trace  on  the  pattern 

the  lines  of  intersection  with  the  upright  prism. 
The  line  of  the  top  base  is  n"  from  the  vertical 
axis  of  the  plate,  and  K  L  is  at  ^=4.25".  The 
true  lengths  of  the  edges  are  found  in  the  projec- 
tions on  V. 


OBLIQUE  INTERSECTION 


PLATE   19. 


59 


60 


MECHANICAL   DRAWING. 


PLATE  20. 


43.  The  Pyramid. — Draw  the  projections  on  H  and  V 
of  a  pyramid  in  three  different  positions,  with  dimen- 
sions as  shown  in  (a). 

(a)  When  the  base  is  parallel  to  H,  and  V  is  parallel 

to  a  side  of  the  base. 
In  the  projection  on  H,  V^  is  at  (2",  6"). 
In  the  projection  on  V,  V0  is  at  (2",  4.5")- 

(b)  When  the  base  is  parallel  to  H,  and  V  makes  an 

angle  of  30  degrees  with  a  side  of  the  base.     Draw 
the  projection  on  H  first. 
In  the  projection  on  H,  V*  is  at  (6",  6"). 


In  the  projection  on  V,  V0  is  at  (6"  ,4.5"). 
(c)  When  the  plane  of  the  base  makes  an  angle  of  30 
degrees  with  H,  and  V  is  parallel  to  a  side  of  the 
base  and  also  parallel  to  the  axis.     Draw  the 
projection  on  V  first. 

In  the  projection  on  H,  Vh  is  at  (11.5",  6"). 

In     the     projection     on     V,    V0    is    at    (11.5", 

4.25"). 

BACA=the  side  of  the  base,  because  the  side  is 
parallel  to  H  and  is,  therefore,  projected  in  its 
true  length. 


PYRAMID 


PLATE  20. 


61 


MECHANICAL   DRAWING. 


PLATE  21. 


44.  Section  and  Development  of  Pyramid. — Draw  the 
projections  of  a  pyramid  3"  high,  with  a  square  base 
whose  side  is  2";  cut  the  pyramid  by  a  section  plane,  show 
the  section,  make  a  projection  on  an  S  plane  parallel  to 
the  section,  develop  the  pyramid,  and  trace  on  the  pat- 
tern the  lines  of  intersection  of  the  pyramid  and  section 
plane. 

(a)  The  base  of  the  pyramid  is  parallel  to  H,  and  V  is 
inclined  to  a  side  of  the  base  at  an  angle  of  30 
degrees. 
In  the  projection  on  H,  the  center  of  the  base  is  at 

(6-75",  6"). 
In  the  projection  on  V,  the  middle  of  the  base  is  at 

(6.75",  1.25"). 
The  section  plane  is  perpendicular  to  H,  and  makes 

an  angle  of  45  degrees  with  V,  Q,  VA  =  £•". 
To  find  CB  and  D,,,  the  projections  on  V  of  the 
points  where  the  edges  are  cut  by  the  section 
plane,  Ck  and  D&  may  be  projected  down  to  Ce 
and  Dv  respectively.  Because  the  angle  formed 
by  the  projecting  line  with  the  projection  of  the 
edge  is  so  small  as  to  make  the  intersection  not 
sharp  or  definite,  another  and  better  method 
is  to  represent  an  auxiliary  or  supplementary 
projection  on  a  plane  parallel  to  both  the 
axis  and  a  diagonal  of  the  base.  One-half  of 


such  a  projection  is  shown  at  (d).  It  is  a  right 
triangle  one  side  of  which,  /  m,  is  one-half  the 
diagonal  of  the  base,  Vh  Gh,  and  the  other 
side  the  altitude  of  the  pyramid.  /  v  is  at  x  =  10", 
and  f m  at  y=4-5".  The  distance  fl=VkCh, 
then  /Q  =  the  perpendicular  distance  from  CB  to 
the  projection  of  the  base  A.v  Ev;  /^  =  VADfc, 
and  k  T)s  =  the  perpendicular  distance  from  Dr 
to  the  projection  of  the  base. 

(b)  Draw  the  projection  on  an  S  plane  that  is  parallel 

to  the  section  plane  to  show  the  section  in  its 
true  size  and  form. 

In  the  projection  on  S,  V,  is  at  (8.5",  4.25"). 

Aj  Ej  is  parallel  to  A/,  E*.  The  perpendicular  dis- 
tance from  Cj  to  the  base  As  E.s  is  equal  to  I  Cs 
in  (d),  and  the  perpendicular  from  Ds  to  A;  E, 
=  £D,  in  (d). 

(c)  In  the  development,  V  is  at  (4.25",  4.25"). 

With  V  as  a  center  and  the  true  length  of  an  edge 
of  the  pyramid  as  a  radius  an  arc  is  described. 
On  this  arc,  each  way  from  m,  the  sides  of  the 
base  are  marked  off  as  chords  of  the  arc. 
AN=AANA;  EG=EAGA;  VC  =  vC,  in  (d), 
which  is  the  true  length  of  the  edge  from  the 
apex  to  the  point  where  the  edge  is  cut  off  by 
the  section  plane  at  C,  and  V  D=w  Ds. 


SECTION  AND  DEVELOPMENT  OF  PYRAMID 


PLATE  21. 


/    "1-    ir     m 


63 


64 


MECHANICAL   DRAWING. 


PLATE  22. 


45.  Hexagonal  Pyramid. — Draw  the  projections  and 
development  of  a  hexagonal  pyramid  with  dimensions 
as  shown. 

(a)  The  base  is  parallel  to  H,  and  V  is  parallel  to  a 

side  of  the  base. 
In  the  projection  on  H,  the  center  of  the  base  is  at 

(2-75",  6"). 
In  the  projection  on  V,  the  center  of  the  base  is  at 

(2.75",  1-25"). 

A  section  plane,  which  intersects  the  axis  in  the 
projection  on  V  at  (2.75",  3.25"),  is  passed  per- 
pendicular to  V  and  makes  an  angle  of  45  degrees 
with  H.  Show  the  section  in  the  projection  on  H. 


(b)  Draw  a  projection  on  P  to  the  right. 

In  the  projection  on  P,  the  center  of  the  base  is  at 
(6.25",  1.25").  Show  the  section  on  this  projec- 
tion. 

(c)  Develop  the  lower  part  of  the  pyramid  below  the 

section. 

The  true  length  of  the  radius  of  the  arc  is  equal 
to  Vv  E,v,  since  this  edge  of  the  pyramid  is  pro- 
jected on  V  in  its  true  length,  being  parallel  to  V. 

The  distances  V  D,  V  C,  etc.,  are  equal  to  the  true 
length  of  the  edges  from  the  apex  to  the  points 
where  they  are  cut  by  the  section  plane,  that 
is,VD=V.D.,VC=V./,etc. 


PLATE  22. 


HEXAGONAL  PYRAMID 


65 


66 


MECHANICAL    DRAWING. 


PLATE  23. 


46.  Intersection  of  Pyramid  and  Prism. — With  dimen- 
sions as  shown,  draw  the  projections  of  a  hexagonal 
pyramid  and  a  square  prism  intersecting;  develop  the 
pyramid,  and  trace  on  the  pattern  the  lines  of  intersec- 
tion of  the  two  surfaces. 

(a)  The  pyramid  has  the  same  position  as  in  the  pre- 
vious plate.  The  prism  has  a  diagonal  of  its  base 
parallel  to  H,  and  the  diagonal  is  equal  to  the 
edge  of  the  base  of  the  pyramid.  V  is  parallel 
to  the  plane  of  its  base. 
In  the  projection  on  H,  the  center  of  the  base  is  at 

(2.75",  4.5"). 
In  the  projection  on  V,  the  center  of  the  base  is  at 

(2.75",  2.5"). 
Draw  the  projection  of  the  prism  first  on  V,  then 

onH. 

BA  and  Dh  are  found  by  projecting  up  from  BB 
and  D,,  respectively,  or  the  distance  from  Bk  to 
the  axial  line  =  the  distance  from  B^  to  the  axial 
line,  etc.  Ch  is  found  as  follows:  Conceive  a 
plane  to  be  passed  parallel  to  the  base  of  the 
pyramid,  cutting  from  its  surface  a  hexagon 
similar  to  the  base.  In  the  projection  on  V, 
F0  is  the  point  where  this  plane  cuts  one  edge 
of  the  pyramid,  Fh  is  its  projection  on  H.  A  line 


drawn  through  Fh  parallel  to  the  side  of  the  base 
is  the  projection  on  H  of  the  side  of  the  hexagon 
cut  from  that  face  by  the  auxiliary  plane. 
The  point  where  the  edge  of  the  prism  meets  the 
projection  of  that  line  on  H  is  Q,.  Similarly  Afc 
and  EA  may  be  found. 

(b)  Draw  the  projection  of  these  two  surfaces  on  P  to 

the  right. 

In  the  projection  on  P,  the  center  of  the  base  of  the 
pyramid  is  at  (6.25",  1.25"). 

CfGf  =  CkGk.  In  the  projection  on  H,  AA-and 
F-h  may  be  found  more  conveniently  and  accu- 
rately from  this  projection  than  by  the  method 
given  above,  thus:  AAKA  =  APKP  and  F.h  Kfc= 
E,K/. 

(c)  In  the  development  of  the  pyramid,  V  is  at  (11.5", 

4.25").  The  front  face  is  placed  symmetrically 
with  the  horizontal  line  in  the  middle  of  the 
plate. 

V  F=VB  F,,,  C  F  is  parallel  to  the  line  of  the  base 
of  that  face  and  equals  Ch  Fft;  A  V  —  Af  Vf, 
EV=EPV,. 

B  V  =  b  VB  and  D  V=rf  V0  the  true  lengths,  re- 
spectively, of  the  edges  of  the  pyramid  from  the 
apex  to  the  points  B  and  D. 


PLATE  23. 


INTERSECTION  OF  PY.RAMID  AND  PRISM 


67 


68 


MECHANICAL  DRAWING. 


PLATE  24. 


47.  The  Cylinder. — Draw  the  projections  of  a  right 
cylinder.  The  radius  of  the  base  is  i"  and  the  height  of 
the  cylinder  is  3". 

(a)  When  the  axis  is  perpendicular  to  H. 

In  the  projection  on  H,  the  center  of  the  base  is  at 

(2",  6"). 
In  the  projection  on  V,  the  center  of  the  lower 

base  is  at  (2",  i"). 

(b)  When  the  axis  makes  an  angle  of   30  degrees  with 

H,  and  V  is  parallel  to  the  axis. 
In  the  projection  on  H,  the  center  of  the  lower 

base  is  at  (4.5",  6"). 
In  the  projection  on  V,  the  center  of  the  lower 

base  is  at  (4.5",  1.75")- 

Draw  first  the  projection  on  V,  then  a  projection 
on  a  supplementary  plane  which  is  parallel  to 

the  base  and  one-quarter  of  an  inch  from  it. 

Show  one-half  of  the  projection  of  the  base. 
To  obtain  the  projections  on  H  of  the  bases,  points 

on  the  line  Ev  Ev  are  projected  up  to  the  line 


DfcC*.    then   CfcB*=C,B,,    DhEh  =  DsEs,    etc.; 
or  the  two  axes  of  the  ellipse  of  the  base  being 
known,  the  ellipse  may  be  constructed  on  the 
axes.     The  bases  are  equal  in  their  projections. 
(c)  Where  the  axis  is  perpendicular  to  H. 

In  the  projection  on  H,  the  center  of  the  base  is  at 

(975",  6"). 
In  the  projection  on  V,  the  center  of  the  base  is  at 

(9-75",  i"). 

A  section  plane  is  passed  cutting  the  axis  in  the 
projection  on  V  at  Ap  (9.75",  2"),  making  an 
angle  of  30  degrees  with  H,  and  is  perpendicular 
to  V.  Show  the  section  on  the  projection  on  H. 

Draw  on  a  supplemental  plane  a  projection  of  the 
ellipse  cut  from  the  surface  by  the  section  plane, 
so  as  to  show  the  section  in  its  true  size  and 
form.  The  supplementary  plane  is  parallel  to 
the  section  plane  and  1.25"  from  it. 

A,  C,  =  AACfc,  BsDs  =  BHDh,  etc.,  or  the  ellipse 
may  be  constructed  on  its  major  and  minor  axes. 


THE  CYLINDER 


PLATE  24 


70 


MECHANICAL   DRAWING. 


PLATE  25. 


48.  Section  and  Development  of  Cylinder  (A). — Draw  the 
section  of  a  right  cylinder  and  develop  it,  tracing  on  the 
pattern  the  outline  of  the  section.  The  diameter  of  the 
base  is  2."  and  the  height  of  the  cylinder  3". 

(a)  The  axis  of  the  cylinder  is  parallel  to  both  H  and  V. 
In  the  projection  on  H,  the  lower  end  of  the 
vertical  diameter  of  the  left  base,  Fk,  is  at 

(i.5",  5-5"). 

In  the  projection  on  V,  Fe  is  at  (1.5",  1.5"). 
Project  one-half  of  the  right  base  on  an  S  plan, 

which  is  parallel  to  this  base.    C,  is  at  (4.75", 

2-5"). 

Pass  a  section  plan  cutting  the  axis  in  the  projec- 
tion on  V  at  a  point  3.8"  from  the  vertical  axis 
of  the  plate,  that  is,  at  Cr.  The  section  plane  is 
perpendicular  to  V  and  makes  an  angle  of  45 
degrees  with  H.  Show  the  section  in  the  pro- 
jection on  H.  Ah  Q,  =  Aj  Cs,  Bh  DA  =  B,  Ds,  etc. 

Such  a  section  will  always  cut  the  whole  or  part  of 
the  curve  of  an  ellipse  (in  this  instance  an  arc  of 
a  circle,  the  circle  being  one  limit  of  the  ellipse) 
from  the  surface  of  the  cylinder.  AA  Q,  =  As  Cs 


=  one-half  the  minor  axis  of  the  ellipse,  and 
H;,Gj,  =  the  projection  of  HB  GB  on  a  horizontal 
line.  Instead  of  determining  the  points  on  the 
curve  of  the  ellipse  by  the  method  indicated,  the 
ellipse  may  be  constructed  on  its  axes. 
(b)  The  cylinder  when  developed  will  be  *a  rectangle, 
the  longer  side  equal  to  the  rectified  *  base,  and 
the  shorter  side  equal  to  the  height  of  the  cylin- 
der. The  point  E  of  the  pattern  is  at  (n",  4.25"). 
To  trace  the  outline  of  the  section  on  the  pattern, 
the  distance  EG  =  the  arc  E,s  Gs  rectified.  The 
divisions  of  the  line  E  G,  produced  each  way 
from  E,  are  equal  to  the  corresponding  divisions 
on  the  semi-circumference  from  Gs  to  Ks  recti- 
fied. AL  =  A1)LB,  or  AR  =  Avr,  BM-B.M.,  or 
=  B  t,  etc. 


*  To  rectify  an  arc  or  curve  is  to  lay  out  on  a  straight  line  a 
distance  equal  to  the -length  of  the  arc  or  curve.  Divide  the  curve 
into  spaces  so  small  that  the  length  of  the  curve  and  its  chord  for 
each  space  shall  not  differ  materially  in  length.  Lay  off  on  the 
straight  line  in  succession  spaces  of  the  same  length  as  those  on 
the  curv£> 


PLATE  25. 


SECTION  AND  DEVELOPMENT  OF  CYLINDER 


71 


72 


MECHANICAL   DRAWING. 


PLATE  26. 


49.  Section  and  Development  of  Cylinder  (B).— Draw  the 
projections  of  a  right  cylinder  whose  height  is  3",  and 
the  diameter  of  whose  base  is  2",  cut  it  by  a  section 
plane,  develop  the  cylinder,  and  trace  on  the  pattern  the 
outline  of  the  section. 

(a)  The  axis  of  the  cylinder  makes  an  angle  of  30  de- 
grees with  H,  and  V  is  parallel  to  the  axis. 
In  the  projection  on  H,  the  center  of  the  base,  Gj, 

is  at  (z",  6"). 
In  the  projection  on  V,  the  center  of  the  base,  Gr, 

is  at  (2",  1.75"). 

Draw  the  projection  on  V  first,  then  find  the  pro- 

^.     jections  of  the  bases  on  H  as  in  (b)  of  Plate  24. 

Pass  a  section  plane  parallel  to  H,   the  edge  of 

which  in  the  projection  on  V  is  2.75"  above  the 

horizontal  axis  of  the  plate.     Show  the  section  in 

the  projection  on  H.     To  do  this,  draw  one-half 

of  the  projection  of  the  cylinder  on  an  S  plane 


parallel  to  the  right  base  and  one-fourth  of  an 
inch  from  it.  A/,  LA  =  twice  a  As,  Eh  Hh  =  C,  Es, 
FhMh  =  fFs,  etc. 

(b)  In  the  development  of  the  cylinder,  let  the  line 
D  B,  which  is  equal  in  length  to  Dv  Ev,  the 
height  of  the  cylinder,  be  4.25"  above,  and 
parallel  to,  the  horizontal  axis  of  the  plate.  Let 
B  be  at  #  =  n".  As  in  the  previous  plate,  the 
pattern  is  a  rectangle,  one  side  of  which  is  equal 
to  the  height  of  the  cylinder  and  the  other  side 
equal  to  its  base  rectified.  A  B  =  arc  As  Bs  rec- 
tified, and  the  space  on  the  line  A  /,  produced, 
are  equal  respectively  to  the  corresponding 
spaces  on  the  arc  A.s  Es  F,  rectified.  G  E  = 
G..E,,,  or  E«  =  E,K.,  FN  =  F.NB,  or  F/= 
F,  QB>  UR  =  U0R0.  The  two  parts  of  the 
pattern  on  either  side  of  the  line  B  D  are 
equal. 


PLATE  26. 


U     R 


73 


74 


MECHANICAL   DRAWING. 


PLATE  27. 


50.  Intersection  of  Prism  and  Cylinder. — Draw  the  pro- 
jections of  a  square  prism  and  a  right  cylinder  which  in- 
tersect and  develop  each,  tracing  on  the  patterns  the 
outline  of  the  intersection. 

The  cylinder  is  3"  in  height,  and  the  diameter  of  its 
base  is  2".  The  diagonal  of  the  base  of  the  prism 
is  1.5". 

(a)  The  axis  of  the  cylinder  is  perpendicular  to  H,  and 

the  axis  of  the  prism  is  perpendicular  to  V.     The 

axes  intersect. 
In  the  projection  on  H,  the  center  of  the  upper 

base  of  the  cylinder  is  at  (2.5",  6"). 
In  the  projection  on  V,  the  center  of  the  upper 

base  of  the  cylinder  is  at  (2.5",  4"). 
In  the  projection  on  H,  the  center  of  the  base  of 

the  prism  is  at  (2.5",  4.5")- 
In  the  projection  on  V,  the  center  of  the  base  of 

the  prism  is  at  (2.5",  3"). 

(b)  Draw  the  projections  of  the  two  surfaces  on  the 

P  plane  to  the  right. 
In  the  projection  on  P,  the  center  of  the  upper  base 

of  the  cylinder  is  at  (6",  4.5"). 
In  the  projection  on  P,  the  center  of  the  base  of 

the  prism  is  at  (4.5",  3"). 
Dp  GP  =  Gh  Dh,    or    Lp  Dp  =  L,,  D,,,    CP  Ep  =  Q,  E,,, 


or 


N.F*=1 


or 


N, 


(c)  Develop  the  front  half  of  the  cylinder,  tracing  on 
the  pattern  the  intersection  of  the  prism  with  it. 

The  middle  line,  C  U,  is  parallel  to,  and  2.25" 
from,  the  horizontal  axis  of  the  plate.  The  left 
side  is  parallel  to,  and  8"  from,  the  vertical  axis 
of  the  plate. 

A  B  =  one-half  the  circumference  of  the  base  of  the 
cylinder  rectified;  the  other  side  of  the  pattern 
equals  the  height  of  the  cylinder. 

B  R  =  BA  Dh  rectified;  the  spaces  on  B  R  produced 
are  equal  respectively  to  the  corresponding,  rec- 
tified spaces  on  the  arc  DA  NA  CA.  R  D  =  Ru  Dv, 


QN=QrNu,  etc.     NS-N.S, 


=  C,,U0,  etc. 


The   part  below  the  middle  line  of  the  pattern 
is  equal  to  the  part  above. 

(d)  Develop  the  prism  and  trace  on  the  pattern  its 
intersection  with  the  cylinder.  The  lower  line  of 
the  pattern  is  parallel  to,  and  5.5"  from,  the 
horizontal  axis  of  the  plate,  and  the  middle  line, 
C  M,  is  8"  from  the  vertical  axis  of  the  plate. 
ML  =  CBD,,  MP  =  C,NC,  etc.,  MC  =  MACA  = 
M,  Cf,  D  L  =  DA  Lh  =  Vf  L,,  N  P  =  NA  PA=N  P 
etc. 


,, 


INTERSECTION  OF  PRISM  AND  CYLINDER 


PLATE  27. 


PP 
If 


>D1r--Gp 


© 


75 


76 


MECHANICAL   DRAWING. 


PLATE  28. 


51.  Intersection  of  Two  Cylinders.  —  Draw  the  projec- 
tions of  two  intersecting  right  cylinders,  with  their  axes 
intersecting  at  right  angles;  develop  each,  and  trace  on 
the  patterns  of  each  its  intersection  with  the  other. 

(a)  The  axis  of  one  cylinder  (for  convenience  called  A) 

is  perpendicular  to  H,  it  is  3"  long,  and  the  base 

of  the  cylinder  has  its  diameter  2".     The  axis  of 

the  other  cylinder  (B)  is  perpendicular  to  V,  and 

the  diameter  of  its  base  is  1.5". 
In  the  projection  on  H,  the  center  of  the  upper 

base  of  the  A  cylinder  is  at  (2.5",  6"). 
In  the  projection  on  V,  the  center  of  the  upper 

base  is  at  (2.5",  4"). 
In  the  projection  on  H,  the  center  of  the  base  of  the 

B  cylinder  is  at  (2.5",  4.5"). 
In  the  projection  on  V,  the  center  of  the  base  is  at 

(2-5",  3"). 

(b)  Draw  the  projection  of  these  cylinders  on  a  P  plane 

to  the  right. 
In  the  projection  on  P,  the  center  of  the  upper 

base  of  the  A  cylinder  is  at  (6",  4"). 
In  the  projection  on  P,  the  center  of  the  base  of  the 

B  cylinder  is  at  (4.5",  3"). 
The  arc  of  the  base  of  the  B  cylinder  which  is  pro- 

jected on  P  in  Mp  PP  is  projected  on  V  in  Cv  Nv 

and  on  H  in  MA  Ph. 


or 


etc. 


f  =  Ch  Ek,  or  C,  M,  =  CA 


N 


or 


The  projection  on  P  of  the  lower  part  of  the  curve 
of  intersection  is  the  same  as  the  upper  half 
inverted. 

(c)  Develop  the  front  half  of  the  A  cylinder.     The 

front  half  of  the  upper  base  rectified  lies  in  the 
line  A  B  which  is  parallel  to  the  vertical  axis  of 
the  plate  and  8"  from  it.  The  middle  element 
of  the  cylinder  between  A  and  B  is  parallel  to  the 
horizontal  axis  of  the  plate  and  2.5"  from  it. 
The  divisions  on  the  line  A  B  are  the  corresponding 
rectified  arcs  between  Q,  and  Dh,  that  is,  Q  R 
=  N,  D,  rectified.  R  D  =  R0  D,,  Q  N  =  QB,  N., 
etc.  CV  =  CBVB>  NU  =  N,Ur,  etc.  The  half 
of  the  curve  below  the  line  C  V  is  the  same  as 
that  above  the  line  inverted. 

(d)  Develop  the  B  cylinder.    The  base  is  rectified  on 

the  line  M  L,  produced  each  way,  which  is  parallel 
to  the  horizontal  axis  of  the  plate  and  5.5"  from 
it;  the  middle  line  M  C  is  parallel  to  the 
vertical  axis  of  the  plate  and  8"  from  it. 
The  divisions  of  the  base  between  M  and  E  are 
equal  to  the  corresponding  divisions,  rectified,  on 
the  circle  between  M  and  LK  in  the  projection 
onV.  MC  =  MpCp  =  MACA)  NP  =  N,PP  =  N,,  Ph, 
L  D  =  Lp  Df  =  LA  DA,  etc.  The  curve  to  the  right 
of  D  L  is  the  same  as  that  to  the  left  reversed. 


PLATE  28. 


INTERSECTION  OF  TWO  CYLINDERS 


77 


78 


MECHANICAL  DRAWING. 


PLATE  29. 


52.  Intersection  of  Cone  and  Prism. — Draw  the  projections 
of  a  right  cone,  3"  high,  with  the  diameter  of  its  base  2.5",  in- 
tersecting a  hexagonal  prism  with  the  diagonal  of  its  base  1.25", 
their  axes  intersecting  at  right  angles.  Develop  both  surfaces 
and  trace  on  the  pattern  of  each  its  intersection  with  the  other, 
(a)  The  axis  of  the  cone  is  perpendicular  to  H,  and  the  axis  of 

the  prism  is  perpendicular  to  V. 
In  the  projection  on  H,  the  apex  of  the  cone,  V>,,  is  at 

(2-5",  6"). 

In  the  projection  on  V,  Vv  is  at  (2.5",  4"). 

In  the  projection  on  H,  the  center  of  the  base  of  the  prism 
is  at  (2.5",  4.5"). 

In  the  projection  on  V,  the  center  of  the  base  of  the  prism 
is  at  (2.5",  2.25"). 

Draw  the  projection  of  the  prism  on  V  first. 

Assume  planes  to  be  passed  perpendicular  to  the  axis  of 
the  cone.  These  planes  will  cut  from  the  surface  of  the 
cone  circles,  and  from  the  prism  lines  parallel  to  its 
edges.  For  convenience  of  construction  pass  these 
planes  at  regular  intervals,  dividing  the  line  D0  Ej,,  in  the 
projection  on  V,  into  12  equal  spaces.  With  VA  as  a 
center  and  V;,  Dj,=  D,,  a  as  a  radius  describe  the  arc 
K;,  D;,  produced.  This  arc  is  the  projection  on  H  of  the 
intersection  of  the  cone  with  the  upper  face  of  the  prism ; 
and  Kh,  the  point  where  this  arc  meets  the  edge  of  the 
prism,  Qh  K/,,  in  the  projection  on  H,  is  the  intersection 
of  that  edge  with  the  surface  of  the  cone.  Similarly  the 
arc  FA  Efc  produced,  and  the  point  F^,  are  found,  £„  b 
=Vfc  E^  Divide  Qj,  M;,,  in  the  projection  on  H,  into 
six  equal  parts  (or  project  the  points  K,,,  Gv,  etc.,  up  to 
Qh,  Mfc,  etc.);  also  divide  the  distance  D^  E^  into  12 
equal  spaces,  and  through  the  points  of  division  draw 
arcs  with  V&  as  a  center  (or  obtain  the  lengths  of  the 
several  radii  from  the  projection  on  V,  as  d  e,  c  f,  etc.). 
These  arcs  are  the  projections  on  H  of  the  arcs  of  circles 
cut  from  the  cone  by  the  12  planes  passed  perpendicular 
to  the  axis.  The  projections  onVof  these  arcs  lie  in  the 
straight  lines  which  would,  if  drawn,  be  perpendicular  to 
the  line  Dj,  Ev  and  divide  it  into  12  equal  spaces. 


In  the  projection  on  H  where  these  arcs  meet,  the  lines  cut 
from  the  prism  by  each  plane  respectively  will  give 
points  on  the  curved  lines  of  intersection  K;,  G^  and 
GhFh. 

(b)  Draw  the  projection  of  these  surfaces  on  a  P  plane  to  the 

right. 
In  the  projection  on  P,  the  apex  of  the  cone,  Vp,  is  at 

(6",  4"). 

QP  DP=U>  Dh,  or  Dp  g'=\h  Dfc;  QP  Kp=Qj,  Kfc,  or 
KP  /=KA  g;  SP  RP=SA  Rfc,  or  RP  n'  =  Rh  n,  etc. 

(c)  In  developing  the  cone  the  apex,  V,  is  at  (8",  4.25"),  VB 

is  horizontal,  the  arc  A  B,  produced,  is  drawn  with  a 
radius  equal  to  VB  A^,  with  the  center  at  V.  Its  length 
equals  the  rectified  circumference  AJ,  B^  A^.  With  radii 
equal  to  Ve  a,  Vv  c,  Vv  f,  etc.,  respectively,  draw 
13  concentric  arcs  in  the  pattern.  These  arcs  are 
the  developed  arcs  cut  from  the  surface  of  the  cone  by 
the  planes  passed  perpendicular  to  its  axis,  as  they  are 
measured  off  in  their  true  lengths. 

Lay  out  on  the  arc  D  K  the  length  of  the  arc  D^  K^  on 
either  side  of  the  center  line  D  E,  and  on  E  F  the  length 
of  Eft  Ffc.  Similarly  lay  off  on  either  side  of  the  center 
line  D  E,  on  the  several  arcs,  the  lengths  of  the  corre- 
sponding arcs  in  the  projection  on  H  respectively. 

(d)  In  developing  one-half  of  the  prism,  lay  out  the  sides 

comprising  half  the  base  on  a  line  parallel  to  the  hori- 
zontal axis  of  the  plate  and  5.5"  from  it.  The  middle 
line,  D  L,  of  the  lower  face  is  parallel  to  the  vertical 
axis  of  the  plate  and  5"  from  it. 

L  D  =  QP  E>f,  g"  D=gt  D?=Vj  DA.  With  g"  as  a  center 
and  V/,  DA  as  a  radius  describe  the  arc  D  K. 

Q  K=QP  KP,  S  R  =  S,  Rf,  etc. 

Eh"=Eph',orVhEh. 

With  h"  as  a  center  and  V;,  E^  as  a  radius  draw  the  arc 
EF.  EL=EpLp=EhLh,FN=EpFp,  etc. 

One-half  the  pattern  of  the  prism  is  shown;  the  other  half 
is  the  same  figure  reversed  on  the  other  side  of  either  of 
the  lines  E  L  or  D  L. 


PLATE  29. 


INTERSECTION  OF  CONE  AND  PRISM 


79 


80 


MECHANICAL  DRAWING. 


PLATE  30. 


53.  Cone  and  Oblique  Cylinder. — Draw  the  projections  of  a  right 
cone,  with  its  axis  vertical,  intersected  by  a  cylinder  whose  right  section 
is  a  circle,  and  whose  axis  is  parallel  to  P  and  makes  an  angle  of  45 
de  rrees  with  both  H  and  V.  The  height  of  the  cone  is  3.25",  and 
the  diameter  of  its  base  2.5".  The  diameter  of  the  base  of  the  cylinder 
is  1.5". 

(a)  In  the  projection  on  H,  the  apex  of  the  cone  VA  is  at  (6.25", 

6.25"). 

Tn  the  projection  on  V,  Vn  is  at  (6.25",  4.5"). 

After  drawing  the  projections  of  the  cone  on  H  and  V,  next 
draw  its  projections  on  P,  and  also  draw  the  base  and  the 
axial  line  of  the  cylinder  on  P. 
(6)  In  the  projection  on  P,  Vf  is  at  (10",  4.5"). 

In  the  projection  on  P,  the  center  of  the  base  of  the  cylinder 
is  at  (9",  3"),  and  the  axial  line  makes  an  angle  of  45  degrees 
with  a  horizontal  line. 

Project  the  base  of  the  cylinder  on  a  supplemental  plane  parallel 
to  it  and  0.25"  from  it,  and  draw  the  outside  elements  of  the 
cylinder  on  P  indefinitely  prolonged  to  the  right. 

Now  if  horizontal  planes  are  assumed  to  be  passed  through 
the  cone  and  cylinder,  they  will  cut  unequal  circles  from  the 
cone  and  equal  ellipses  (in  whole  or  in  part)  from  the  cylinder. 
Pass  12  or  more  of  these  planes  equally  distant  apart.  If  the 
cylinder  is  supposed  to  be  extended  indefinitely  to  the  right  of 
the  base,  the  cutting  plane  whose  edge  is  Ep  KP  will  cut  the 
ellipse  from  the  cylinder  shown  at  (d),  the  longer  axis  of  which  = 
Ep  Kp,  and  the  shorter  axis  equals  the  diameter  of  the  base 
of  the  cylinder.  The  center  of  this  ellipse  is  at  (9.3",  6"). 

The  plane  whose  edge  is  C^,  5  cuts  from  the  cone  a  circle  whose 
projection  on  H  is  an  equal  circle  whose  radius  is  VA  s' =u  s 
in  the  projection  on  P.  This  same  plane  cuts  from  the  ex- 
tended cylinder  a  part  of  an  ellipse  whose  projection  on  H  is 
equal  to  itself,  one  side  of  which  is  V  Ch.  Where  the  circle 
and  ellipse  meet,  at  Ch,  is  a  point  lying  on  the  surface  of  both 
the  cone  and  cylinder  and  is  therefore  a  point  in  their  inter- 
section. Similarly  the  other  points  in  the  projection  on  H 
are  found,  as  LA,  FA,  MA,  etc. 

If  the  ellipse  Es  Kj  be  traced  on  vellum  cloth  and  accurately 
cut  out,  the  points  where  the  ellipse  meets  the  respective 


circles  in  LA,  Ch,  FA,  etc.,  may  be  found  in  the  following  man- 
ner: For  Ch  lay  off  on  the  vertical  axial  line  drawn  through 
Vh  the  distance  V*  A'=«  h  in  the  projection  on  P,  place  the 
pattern  of  the  ellipse  so  that  Ks  is  on  h'  and  the  long  diameter, 
Kj  Es,  coincides  with  the  vertical  axial  line,  then  the  edge  of 
the  pattern  meets  the  corresponding  circle  at  Ch.  Vh  m'  —w  m; 
here  K,  is  placed  on  m'  and  the  edge  of  the  pattern  meets 
the  corresponding  circle  at  MA;  the  radius  of  the  correspond- 
ing circle  =iv  r. 

Similarly  the  other  points  may  be  found,  thus  determining  the 
curve  of  intersection  in  the  projection  on  H  in  the  points 
B*.  LA,  Ch,  FA,  MA,  etc. 

In  the  projection  on  P,  Cp  u  is  equal  in  length  to  the  distance 
from  CA  to  the  horizontal  axial  line  in  H,  and  the  distance 
of  Cv  from  the  axial  line  in  V  is  equal  to  the  distance  from 
CA  to  the  vertical  axial  line  in  H.  Similarly  the  other  points 
of  the  curve  of  intersection  on  P  and  V  may  be  found. 

The  projections  on  H  and  V  of  the  outline  of  the  base  of  the 

cylinder  may  be  found  as  in  previous  problems. 
(c)  A  projection  of  these  bodies  is  made  on  an  S  plane  which  is 
perpendicular  to  H  and  makes  an  angle  of  60  degrees  with  V. 

The  line  e  c,  produced,  distant  2.5"  from  VA,  shows  the  position 
of  this  plane.  Instead  of  showing  this  projection  on  e  c  in 
this  position,  it  is  changed  to  another  position  in  line  with  the 
bases  of  the  projections  of  the  cone  in  V  and  P,  in  order  to 
utilize  more  conveniently  the  heights  of  the  several  points 
in  those  projections. 

In  the  projection  on  S  (in  the  second  position),  the  apex  of  the 
cone,  V,,  is  at  (2.5",  4.5"). 

To  find  Mi,  first  project  VA  to  c  and  MA  to  d,  then  on  the  hori- 
zontal line  through  Mj,  lay  off  from  the  axial  line  through  Vs 
the  distance  to  Ms  =d  c;  on  the  horizontal  line  through  Ft, 
the  distance  from  the  axial  line  to  Fj  =b  c,  b  being  the  pro- 
jection on  e  cot  the  point  FA  in  the  projection  on  H.  Similarly 
the  other  points  of  the  line  of  intersection  as  projected  on  the 
S  plane  are  found,  as  Bj,  Lj,  Cj,  Fs,  etc. 

The  points  on  the  outline  of  the  base  of  the  cylinder  are  found 
in  a  similar  way  to  that  in  which  the  points  oi1  the  line  of  inter- 
section are  found. 


CONE  AND  OBLIQUE  CYLINDER 


PLATE  30. 


\ 


81 


CHAPTER  V. 


ISOMETRIC    AND  OBLIQUE   PROJECTIONS,   SHADES,   SHADOWS,  AND  PERSPECTIVE. 


54.  Isometric  Projection. — A  projection  of  an  object 
may  be  made  on  one  plane  to  show  three  dimensions  (see 
Plate  11,  (c)).  In  order  that  such  a  drawing  may  be 
useful  as  a  working  drawing,  the  lengths  and  true  forms 
of  the  lines  must  be  known.  In  an  isometric  drawing 
the  principal  lines,  although  not  parallel  to  the  plane  of 
projection,  are  shown  in  their  true  lengths.  If  the  plane 
of  projection,  or  plane  of  the  paper,  is  taken  perpendicu- 
lar to  a  body  diagonal  of  a  cube,  the  three  adjacent 
edges  of  the  corner  to  which  the  diagonal  is  drawn  will 
be  projected  on  this  plane  in  lines  which  make  equal 
angles  with  each  other,  viz.,  angles  of  120  degrees. 
These  lines  are  called  the  isometric  axes  of  an  isometric 
projection.  One  of  the  three  axes  is  usually  made 
vertical. 

Since  lines  that  are  parallel  to  each  other  are  projected 
in  parallel  lines,  if  the  principal  lines  of  an  object  are 
parallel  to  the  three  edges  of  a  cube  (which  form  a  right 


trihedral  angle),  they  may  be  projected  parallel  to  the 
isometric  axis. 

The  inclination  to  the  plane  of  projection  of  the  edges 
of  the  cube  is  35  degrees  16  minutes  nearly,  then  (by 
trigonometry)  the  projections  of  any  definite  portion  of 
an  edge,  or  any  line  parallel  to  it,  will  be  equal  to  the 
length  of  that  portion  multiplied  by  the  cosine  of  35 
degrees  16  minutes.  An  isometric  scale  may  thus  be 
made  by  taking  any  unit  of  measure  on  the  object  and 
making  the  unit  of  measure  on  the  drawing  equal  to  the 
cosine  of  35  degrees  16  minutes  times  that  unit.  But  an 
isometric  scale  is  quite  unnecessary,  since  all  lines  which 
are  projected  parallel  to  the  isometric  axes  are  equally 
foreshortened.  An  ordinary  scale  may  be  used. 

Shade-lines. — In  isometric  projections  the  rays  of  light 
are  assumed  parallel  to  the  plane  of  projection  and  making 
with  the  horizontal  an  angle  of  30  degrees;  the  shade-lines 
are  determined  accordingly. 

82 


84 


MECHANICAL   DRAWING. 


55- 


PLATE  31. 


(a)  Draw  the  isometric  projection  of  a  cube  whose  edge 
is  2".  o  x,  o  y,  and  o  z  are  taken  as  the  isometric 
axes;  these  lines,  each  2"  long,  may  be  taken  as 
the  projections  of  the  three  adjacent  edges  of  the 
cube,  o  x  and  o  y  each  make  an  angle  of  30  degrees 
with  the  horizontal.  (Use  the  T  square  and  30-60 
degree  triangle.)  In  (a),  o  is  at  (3",  4.25"). 

(d)  Draw  a  circle  in  the  upper  base  whose  diameter  is 
2".  In  (a),  measure  i"  from  x  to  d,  draw  dc 
parallel  to  o  x.  c  g  is  parallel  to  o  z.  In  (d), 
c  is  at  (6",  6").  Lay  off  eg  in  (a)  =  cg  in  (d). 
Draw  g  h  in  (a)  parallel  to  o  x  and  equal  to 
g  h  in  (d).  Proceed  in  a  similar  manner  to  find 
other  points. 


(b)  Draw  the  isometric  projection  of  a  vertical,  hex- 
agonal prism,  3^"  long,  a  side  of  whose  base  is 
0.75",  intersected  by  a  rectangular  prism  ij" 
long,  the  edges  of  whose  base  are  1.25"  hori- 
zontal and  i"  vertical,  and  which  is  parallel  to 
one  face  of  the  hexagonal  prism.  The  upper 
face  of  the  intersecting  prism  is  i"  below  the 
upper  base  of  the  hexagonal  prism,  ox,  o  y, 
and  o  z  may  be  taken  as  the  axes  or  parallel  to 
them.  In  (b),  c  is  at  (9",  6").  a  b  in  (b)  is  par- 
allel to  o  x,  and  h  k  parallel  to  o  z.  c  k  =  c  h  in 
(b)  equals  ck  =  ch  in  (c).  ez  in  (b)  =  i".  eg 
in  (b)  =  e  g  in  (c),  etc.  In  (c),  c  is  at  (6", 
2.25"). 


ISOMETRIC  PROJECTION 


PLATE  31. 


85 


86 


MECHANICAL    DRAWING. 


56.  Oblique    Projections.  —  All    the   preceding    projec- 
tions have  been  orthographic,  that  is,  the  projecting  lines 
of  the  different  points  have  been  perpendicular  to  the 
plane  upon  which  the  drawing  is  made.     //  the  projecting 
lines  are  not  perpendicular  to  this  plane,  though  parallel 
to  each  other,  the  projections  are  called  oblique. 

57.  Cavalier   Projections.  —  When   the   projecting   lines 
make  an  angle  of  45  degrees  with  the  plane  of  projection, 
in  any  direction,  the  projection  is  oblique  and  is  called  a 
cavalier  projection. 

Cavalier  projections  are  applicable,  in  general,  to 
simple  objects  or  bodies  whose  faces  are  at  right  angles 
to  each  other.  One  face  is  usually  taken  parallel  to  the 
plane  of  projection  and  so  is  shown  in  its  true  size;  also 
the  edges  that  are  perpendicular  to  the  plane  of  projec- 
tion are  shown  in  their  true  lengths,  although  the  faces 
they  bound  are  not  shown  in  their  true  shape  or 
size. 

The  advantage  of  cavalier  projections  for  a  working 
drawing  is  that  the  three  dimensions  are  shown  in  one 
projection  and  the  principal  lines  (those  parallel  and 


perpendicular  to  the  plane  of  projection)  are  shown  in 
their  true  lengths.- 

58.  Pseudo-perspective.  —  When  the  projecting  lines 
make  any  angle  less  than  45  degrees  with  the  plane  of  pro- 
jection, in  any  direction,  the  projection  is  oblique  and 
is  called  pseudo-perspective.  It  resembles  somewhat  a 
true  perspective  which  will  be  explained  later.  As  in 
isometric  and  cavalier  projections  its  application  is  lim- 
ited to  the  projections  of  bodies  made  up  principally  of 
plane  faces  which  are  perpendicular  to  each  other.  One 
face  is  usually  taken  parallel  to  the  plane  of  projection 
and  (together  with  all  other  planes  parallel  to  it )  is  shown 
in  its  true  size,  while  the  projections  of  the  edges  which 
are  perpendicular  to  the  plane  of  projection  are  fore- 
shortened to  any  proportion  of  their  true  length,  as  one- 
half,  two-thirds,  etc.,  by  taking  the  projecting  lines  at  a 
corresponding  angle  to  the  plane  of  projection. 

Shade-lines. — In  oblique  projections  the  rays  of  light 
are  assumed  to  have  the  same  direction  as  in  orthographic 
projections,  viz.,  the  direction  of  the  body  diagonal  of  a 
cube  (see  Art.  34). 


88 


MECHANICAL  DRAWING. 


PLATE  32. 


59.  Draw  the  cavalier  projections  of  a  cube. 

(a)  The  corner  A  is  at  (1.75",  4.25").  The  line  D  d 
makes  an  angle  of  30  degrees  with  the  horizontal 
(it  may  make  any  desired  angle).  The  projec- 
tion of  the  circle  on  the  front  face  is  a  circle 
equal  to  the  circle  it  represents.  On  the  top 
face  c  d  is  divided  into  the  same  number  of 
parts,  as  C  D  on  the  front  face,  then  g  h  =  G  H, 
etc.  A  is  a  point  in  the  circle,  and  other  points 
are  found  in  a  similar  way. 


(b)  Draw  the  pseudo-perspective  of  a  cube.  The 
corner  A  is  at  (8",  4.25").  The  line  D  d  makes 
an  angle  of  30  degrees  with  a  horizontal,  and  in 
this  instance  is  made  i"  long  or  one-half  the 
length  of  the  line  it  represents.  D  d  may  have 
any  desired  direction  and  length,  so  long  as  the 
length  is  less  than  the  line  it  represents.  Points 
in  the  circumference  of  the  circle  on  the  top 
face  may  be  found  similarly  to  those 
in  (a). 


OBLIQUE' PROJECTION 


PLATE  32. 


CAVALIER 


PSEUDO.PERSPECTIVE 


/^"D^ 
/         1G 

r            i 

~r 

1 

"fc 

k^-J»x 

s* 

.JL- 


s--  -     |-tu 


-\H 


90 


MECHANICAL    DRAWING. 


60.  Shades  and  Shadows. — Regarding  the  sun  as  the 
source  of  light  in  determining  shades  and  shadows,  the 
rays  of  light  arc  practically  parallel  and  may  be  so  regarded. 

That  part  of  a  body  which  faces  towards  the  source  of 
light  and  upon  which  the  rays  of  light  fall  is  said  to  be 
in  the  light,  and  that  part  which  faces  away  from  it  and 
upon  which  the  rays  do  not  fall  is  said  to  be  in  the  shade. 

The  division-line  between  the  light  and  shaded  parts  of 
a  body  is  called  the  line  of  shade. 

A  shadow  on  any  surface  is  that  part  of  the  surface 
from  which  the  light  is  excluded  by  the  intervention  of 
some  opaque  body  between  it  and  the  source  of  light. 
Parts  of  a  body  may  cast  shadows  on  other  parts  of  the 
same  body  that  are  not  in  the  shade,  or  upon  other  bodies 
or  surfaces. 

When  one  body  casts  a  shadow  on  another,  the  line  of 
shade  casts  the  outline  of  the  shadow. 

The  difference  between  a  shade  and  a  shadow  is  appar- 


ent. A  shade  is  that  part  of  any  body  or  surface  which 
is  turned  away  from  the  light,  and  a  shadow  is  that  part 
of  a  body  or  surface  from  which  the  light  is  cut  off  by 
some  opaque  body. 

To  find  the  projection  of  the  line  of  shade  of  a  body 
which  is  made  up  of  faces  that  are  planes,  the  method  of 
Art.  34  for  determining  which  faces  are  in  the  light  and 
which  in  the  shade  may  be  followed.  If  the  body  is 
made  up  of  curved  surfaces,  it  will  be  necessary  to  find 
the  projection  of  rays  of  light  which  form  an  enveloping, 
or  tangent,  surface;  the  line  of  tangency  will  be  the  line 
of  shade. 

The  shadow  of  a  point  on  any  surface  is  found  by 
determining  where  a  ray  of  light  through  the  point 
pierces  the  surface;  the  projection  of  the  shadow  of  the 
point  is  found  by  using  the  projections  of  the  ray. 

//  a  line  is  parallel  to  a  plane,  the  shadow  of  the  line  on 
the  plane  will  be  parallel  and  equal  to  the  line. 


92 


MECHANICAL  DRAWING. 


PLATE  33. 


6l.   Find  the  projections  of  the  shadow  of  a  cube  on  a  hori- 
zontal plane,  also  on  a  vertical  plane  which  is  parallel  to  V. 
(a)  Take  the  faces  of  the  cube  parallel  to  both  H  and  V. 

The  horizontal  plane  (T)  upon  which  the  shadow  is  cast 
coincides  with  the  bottom  face  of  the  cube,  the  trace  of 
which  on  Vis  £2  B2.  The  vertical  plane  (R)  is  one-half 
an  inch  back  of  the  cube,  the  trace  of  which  on  H  is  the 
horizontal  line  through  BI. 
The  edge  of  the  cube  is  2".  Ak  is  at  (3",  3.5")  and  AK  is 

at  (3",  3"). 

The  line  of  shade  of  the  cube,  beginning  at  the  bottom 
under  A,  follows  the  edge  to  A,  thence  along  the  top 
edge  to  C,  thence  to  D,  and  thence  to  the  bottom 
under  D. 

The  projection  on  H  of  the  ray  of  light  through  A  is  AA  AI  ; 
its  projection  on  V,  A,,  B2.  This  ray  pierces  the  T  plane 
at  a  point  whose  V  projection  is  B2  and  whose  H  projec- 
tion is  AI.  The  V  projection  of  the  entire  shadow  on  the 
T  plane  will  fall  in  the  line  £3  B2;  the  projection  on  H 
of  the  outline  of  the  shadow  is  A^  At  B!  Q  DI  D&.  The 
line  AI,  A!  is  the  projection  of  the  shadow  of  the  vertical 
edge  of  the  cube  through  A,  the  line  AjCj  that  of  the  edge 
A  C,  Ci  Dj  that  of  the  edge  C  D,  and  Dx  DA  that  of  the 
vertical  edge  through  D. 

That  part  only  of  the  projection  on  V  of  the  shadow  of  the 
cube  on  the  R  plane  which  is  above  the  T  plane  is  shown. 
The  shadow  of  the  vertical  edge  through  A  does  not 
reach  the  R  plane.  That  part  of  the  edge  A  C  from 


B  to  C  casts  the  shadow  which  is  projected  on  V  in  the 
line  B2  C2,  C2  D2  is  the  projection  on  V  of  the  edge  C  D, 
and  D2  E2  of  a  part  of  the  vertical  edge  through  D,  both 
parallel  and  equal  to  edges  casting  the  shadows. 

The  H  projection  of  the  entire  shadow  on  the  R  plane 
falls  in  the  horizontal  line  through  BI.  The  H  projec- 
tion of  the  shadow  of  the  edge  A  C  is  At  C1;  equal  and 
parallel  to  A  C ;  also  Dj  Ci  is  equal  and  parallel  to  D  C. 
(b)  A  face  of  the  cube  is  parallel  to  H,  and  another  makes  an 
angle  of  30  degrees  with  V.  The  T  plane  is  one-half 
an  inch  below,  and  parallel  to,  the  bottom  face  of  the 
cube.  The  trace  of  the  R  plane  on  H  is  the  line  through 
K!  (n",  7")  making  an  angle  of  15  degrees  with  a 
horizontal. 

Eh  is  at  (7.5",  4.25"),  and  E,  at  (7.5",  1.5"). 

The  line  of  shade  follows  in  order  the  edges  passing  through 
the  corners  A',  A,  C,  D,  D',  E',  A'. 

The  outline  of  the  shadow  on  the  T  plane  as  projected  on 
H  is  AI  KI  FI  DI  EI  Aj,  and  the  outline  of  the  shadow 
on  the  R  plane  as  projected  on  V,  F2  K2  A2  C2D2  F2. 

No  part  of  the  shadow  of  the  cube  on  the  T  plane  is  shown 
beyond  the  R  plane,  and  no  part  of  the  shadow  on  the 
R  plane  is  shown  below  the  T  plane. 

The  section  lines  showing  the  projection  of  shadows  should 
be  drawn  horizontal  when  the  shadow  is  on  a  horizon- 
tal plane,  and  vertical  when  the  shadow  is  on  a  vertical 
plane.  The  section-lines  should  be  omitted  when  the 
shadow  is  hidden  by  the  body  in  the  projections. 


PLATE  33. 


SHADOWS 


j\ 

N  I        \ 

ME, 


C," 

.  I 


'B2 


93 


94 


MECHANICAL   DRAWING. 


62. 


PLATE  34. 


(a)  Find  the  projections  of  the  shaded  parts  of  a  cube  and  of 
its  shadow  on  a  horizontal  plane  (T)  and  on  a  vertical  plane 
(R)  parallel  to  V.     Show  only,  that  part  of  the  shadow  on 
the  T  plane  which  is  in  front  of  the  R  plane,  and  that  part 
of  the  shadow  on  the  R  plane  which  is  above  the  T  plane. 

One  edge  of  the  cube  is  parallel  to  H,  the  adjacent  faces  making 

equal  angles  (45  degrees)  with  H,  and  a  face  making  an 

angle  of  30  degrees  with  V. 
Eh  is  at  (2",  4.5")-  and  Ev  at  (2",  0.75"). 
The  T  plane  coincides  with  the  bottom  edge  of  the  cube,  and 

the  R  plane  passes  through  the  corner  E  which  is  farthest 

from  V.  ' 
The  line  of  shade  is  B  A  D  C  E  G  B;   the  projection  of  the 

part  of  its  shadow  shown  on  the  T  plane  is  B^  Aj  DI  FI  GI  Eh, 

and  of  that  part  shown  on  the  R  plane   D2  Cz  Ev  p2  D2. 

G  F  is  the  only  part  of  the  edge  G  E  which  casts  a  shadow 

on  the  T  plane,  and  E  F  the  part  which  casts  a  shadow  on 

the  R  plane. 
The  faces  that  are  in  the  shade  may  be  found  by  the  method 

of  Art.  34.     A  B  D  is  the  only  visible  face  of  the  cube  in 

the    shade. 

(b)  Find  the  projections  of  the  shaded  part  of  a  cylinder,  and 

of  the  shadow  of  the  cylinder  on  a  horizontal  plane  (T) 
coinciding  with  the  lower  base,  and  on  a  vertical  plane  (R) 
parallel  to  V  whose  horizontal  trace  is  at  y=5-JS".  Show 
the  projection  of  the  entire  shadow  on  the  T  plane;  on 
the  R  plane  show  only  that  part  which  is  above  the  T  plane. 
While  it  is  conventional  to  regard  the  projections  of  rays  of 
light  on  H  and  V  as  making  an  angle  of  45  degrees  each 
with  a  horizontal,  nevertheless  it  is  optional  with  the  drafts- 
man to  assume  them  in  any  reasonable  direction.  In  this 
problem  let  the  projections  of  rays  on  H  make  angles  of  30 


degrees,  and  those  on  V  an  angle  of  45  degrees,  with  the 
horizontal. 

The  axis  of  the  cylinder  is  taken  perpendicular  to  H  and 
parallel  to  V. 

Ch  is  at  (8",  4-5"),  and  Cv  at  (8",  3.25"). 

The  line  of  shade  is  D'  D  G  E  E'  K'  D';  the  projection  of 
its  shadow  on  the  T  plane  is  D&  DI  FI  EI  EA  KA  D&,  and 
the  part  of  the  shadow  shown  on  the  R  plane  p2  G2  £2  N2  F2. 
E  N  is  the  only  part  of  the  line  of  shade  E  E'  whose  shadow 
shows  on  the  R  plane. 

The  projection  on  the  T  plane  of  the  outline  of  the  shadow 
DI  FI  EI  is  a  semicircle  with  its  center  at  Ci,  equal  to  the 
semicircle  D  G  E  which  casts  the  shadow.  The  projection 
on  the  R  plane  of  the  shadow  of  the  same  semicircle  is  found 
by  determining  the  projections  of  the  shadows  of  its  several 
points  and  joining  them  by  an  irregular  curve. 

The  shaded  part  of  the  cylinder  is  the  one-half  which  includes 
the  bottom  base  and  that  part  of  the  lateral  surface  away 
from  the  source  of  light  bounded  by  the  line  of  shade,  viz., 
D  D'  F'  G'  E'  E  G  F  D. 

A  few  parallel  shade-lines,  gradually  increasing  in  their  dis- 
tance apart,  are  drawn  at  the  left  of  the  figure  to  give  the 
appearance  of  a  curved  surface,  notwithstanding  that  part 
of  the  surface  is  wholly  in  the  light. 

Note:  The  application  of  the  principles  of  light  and  shade  as 
they  apply  to  the  representation  of  curved  surfaces  will  not 
be  attempted  here;  should  the  student  desire  to  look  farther 
into  the  subject  he  is  referred  to  Randall's  Shades,  Shadows, 
and  Perspective;  J.  E.  Hill's  Shades,  Shadows,  and  Per- 
spective; and  Notes  on  Shades,  Shadows,  and  Perspective, 
by  C.  E.  Crandall,  revised  and  enlarged  by  Walter  L. 
Webb. 


SHADES  AND  SHADOWS 


PLATE  34. 


E*          F, 


95 


96 


MECHANICAL   DRAWING. 


63.  Perspective. — In  the  projections  which  have  pre- 
ceded, the  projecting  lines  have  been  parallel  to  each 
other,  that  is,  the  position  of  the  eye  has  been  assumed 
as  at  an  infinite  distance  from  the  plane  of  projection.  If 
the  position  of  the  eye  is  at  a  finite  distance  from  the  plane 
of  projection,  the  projecting  lines  will  not  be  parallel,  but 
will  meet  the  plane  of  projection  at  different  angles. 
The  points  where  the  projecting  lines  meet  the  plane  of 
projection  determine  the  projection  as  in  preceding  cases. 
Such  a  projection  is  called  a  perspective. 

A  perspective  may  be  imagined  if  one  looks  with  one 
eye  only  at  an  object,  such  as  a  house,  through  a  window, 
and  sees  traced  on  the  glass  the  outlines  of  the  house  a? 
they  are  projected  upon  it. 

A  perspective  is  a  true  representation  of  an  object  as 
the  eye  sees  it,  but  it  is  useless  as  a  working  drawing, 
because  the  projections  of  parallel  and  equal  lines  have  not 
the  same  lengths  except  when  the  lines  they  represent  are 
both  parallel  to  the  plane  of  projection  and  at  the  same 
distance  from  it. 

A  photograph  is  a  perspective,  and  since  an  existing 
object  or  body  may  be  photographed  much  more  cheaply 
and  satisfactorily  (except  for  very  simple  objects)  than 
a  perspective  drawing  can  be  made,  it  is  evident  a  per- 
spective serves  the  purpose  of  representing  non-existing 
bodies,  e.g.,  a  dwelling  about  to  be  erected. 

To  make  a  mathematical  or  accurate  perspective  of  a 
body  which  does  not  exist,  it  is  first  necessary  to  draw 


orthographic  projections  on  H  and  V,  and  locate  the 
orthographic  projections  of  the  position  of  the  eye.  Also 
it  is  necessary  to  deal  with  the  orthographic  projections 
of  the  projecting  lines  rather  than  with  the  lines  them- 
selves. 

The  projecting  lines  are  called  visual  rays. 

The  plane  of  projection  is  called  the  picture-plane. 

The  position  of  the  eye  is  called  the  point  of  sight,  and 
the  orthographic  projection  of  the  point  of  sight  on  the 
picture-plane  is  called  the  center  of  the  picture. 

The  trace  on  the  picture-plane  of  a  horizontal  plane  at 
the  height  of  the  eye  is  called  the  horizon.  The  center  of 
the  picture  is  always  on  the  horizon. 

The  perspective  of  a  point  is  determined  directly  by 
finding  where  the  projecting  line  of  that  point  meets  the 
picture-plane,  or  otherwise  by  finding  the  intersection  of 
the  perspectives  of  two  lines  which  pass  through  the 
point.  The  last  method  is  the  one  often  adopted,  because 
it  does  not  require  the  vertical  projection  of  the  body 
to  be  drawn,  but  uses  a  vertical  line  on  which  are  marked 
off  the  heights  of  the  different  points.  This  line  is 
called  the  line  of  heights. 

The  projections  of  all  lines  which  are  not  parallel  to 
the  picture-plane,  but  which  are  parallel  to  each  other, 
have  one  point  of  perspective  in  common  which  is  called 
the  vanishing-point. 

The  vanishing-point  of  a  system  of  parallel  lines  is  deter- 
mined by  finding  where  a  line  through  the  point  of  sight 


ISOMETRIC   AND    OBLIQUE    PROJECTIONS,  SHADES,  SHADOWS,  AND  PERSPECTIVE. 


97 


parallel  to  the  lines  of  the  system  pierces  the  picture-plane. 
The  proof  of  this  is  as  follows:  Any  one  of  the  parallel 
lines  of  the  system  and  the  line  through  the  point  of 
sight  parallel  to  it  form  a  plane  which  intersects  the 
picture-plane  in  a  line  which  contains  the  perspective  of 
the  one  line.  Any  other  one  of  the  parallel  lines  and  the 
same  line  through  the  point  of  sight  form  another  plane 
the  trace  of  which  on  the  picture-plane  contains  the  per- 
spective of  the  second  line.  Since  both  these  planes  con- 
tain the  parallel  line  through  the  point  of  sight  in  common, 
their  traces  on  the  picture-plane  will  intersect  in  the 
point  where  that  line  pierces  the  picture-plane;  hence 
this  point  will  be  common  to  the  perspectives  of  the  two 
parallel  lines  and  all  other  lines  parallel  to  them. 

The  center  of  the  picture  is  the  vanishing-point  of  all 
lines  perpendicular  to  the  plane  of  the  picture.  Such  lines 
are  called  perpendiculars. 

The  horizon  will  contain  the  vanishing-points  of  all  lines 
that  are  horizontal  whatever  angle  they  may  make  with  the 
picture-plane.  The  vanishing-point  of  horizontal  lines 
which  make  the  angle  of  45  degrees  with  the  picture- 
plane  will  be  in  the  horizon  at  a  distance  from  the  center 
of  the  picture,  equal  to  the  distance  of  the  point  of  sight 
from  the  picture-plane.  The  vanishing-point  of  such 
lines  is  called  the  point  of  distance,  and  the  lines  are  called 
diagonals. 

The  perspectives  of  all  lines  parallel  to  the  picture-plane 
are  parallel  to  the  projections  of  the  lines.  The  vanishing- 


point  of  a  system  of  such  lines  is  at  infinity,  that  is,  their 
perspectives  are  parallel  to  each  other.  All  lines  parallel 
to  the  picture-plane  which  are  vertical  are  called  verticals, 
and  if  horizontal  they  are  called  horizontals. 

The  two  lines  which  are  used  to  determine  the  perspec- 
tive of  a  point  are  in  general  a  perpendicular  and  a  diag- 
onal, or  a  vertical  with  either  of  these  two,  or  a  horizontal 
with  either  of  these  three. 

64.  To  find  the  perspectives  of  shadows  on  any  surface 
in  general,  find  the  perspective  of  the  point  where  the  ray  of 
light  through  the  point  pierces  the  surface.  If  it  is  a  shadow 
on  a  horizontal  plane,  find  the  perspective  of  the  ray  of 
light  and  the  perspective  of  the  projection  of  that  ray 
on  the  plane,  and  where  these  perspectives  meet  will  be 
the  perspective  of  the  shadow  of  the  point. 

If  the  rays  of  light  have  the  direction  as  given  in  Art. 
34,  then  the  projection  of  rays  on  H  being  diagonals, 
their  perspectives  will  vanish  in  the  horizon  at  the  point 
of  distance  (D^),  and  the  vanishing-point  of  the  perspec- 
tive of  rays  will  vanish  at  a  point  in  a  vertical  line  through 
D^  as  far  below  it  as  D^  is  from  the  center  of  the  picture 
(SJ.  But  the  direction  of  the  rays  of  light  may  be 
assumed  at  will  so  long  as  the  conditions  are  reasonable; 
therefore  RA)  somewhere  in  the  horizon,  may  be  taken  as 
the  vanishing-point  of  the  perspectives  of  the  projections 
of  rays  on  a  horizontal  plane,  and  R^  somewhere  directly 
below  as  the  vanishing-point  of  the  perspective  of  the 
rays.  The  nearer  to  the  horizon  RK  is  taken  the  more 
nearly  horizontal  will  be  the  rays  of  light,  and  the  nearer 
Rh  is  to  S0  the  more  nearly  are  the  rays  of  light  behind 
the  draftsman. 


98 


MECHANICAL   DRAWING. 


65- 


PLATE  35. 


Find  the  perspective  of  a   cube,  the  perspective  of  its  shade,  and  the 

perspective  of  its  shadow  on  a  horizontal  plane.     The  edge  of  the 

cube  is  2",  and  faces  are  parallel  to  H  and  V. 
A*  is  at  (3.5",  5"),  and  A.  at  (3.5",  3.5"). 
K*  L*  is  the  trace  on  H  of  the  picture-plane  at  y  =4.75". 
Sv  De  is  the  horizon  at  y  =4.25". 
A,,'  Br',  the  trace  on  the  picture-plane  of  the  horizontal  plane  (T)  upon 

which  the  cube  rests,  is  at  y  =  1.5". 
S^  is  the  center  of  the  picture  at  x =8. 25". 
D,,      is      the      distance-point     (vanishing-point    of    diagonals)     at 

*  =  12". 
RA  is  the  vanishing-point  of  the  projection  of  rays  of  light  on  the  T 

plane  at  x=-io",  and  R^the  vanishing-point  of  rays  of  light  at 

y=°-75" 

The  picture- plane  and  V  are  one  and  the  same  plane. 

Find  the  perspective  of  the  corner  A'  by  a  diagonal  and  a  perpendicular. 
S/  Dv  is  the  perspective  of  the  diagonal,  and  A/  S,,  the  perspective 
of  the  perpendicular.  The  two  perspectives  meet  at  a',  which  is 
the  perspective  of  the  corner  A.  For  the  diagonal  through  A  is 
projected  on  H  in  the  line  A*  S*,  and  on  V  in  the  line  A/  S,,'.  This 
diagonal  meets  the  picture-plane  at  a  point  whose  projection  on  H 
is  S*,  and  on  V  at  S»';  therefore  St>'  is  the  perspective  of  one  point 
of  the  diagonal,  and  the  vanishing-point  of  diagonals  D,,  is  another 
point. 

Find  the  perspective  of  the  corner  A  by  a  vertical  and  a  perpendicular. 
a'  a  is  the  perspective  of  the  vertical  through  A,  and  A,,  Sv  the 
perspective  of  the  perpendicular.  For  the  perspective  of  the  ver- 
tical a'  a  is  parallel  to  the  projection  of  the  vertical  on  the  picture- 
plane;  also  the  perspective  of  the  perpendicular  will  vanish  at 


the  center  of  the  picture,  and  another  point  of  its  perspective  will 
be  where  the  perpendicular  meets  the  picture-plane. 

Find  the  perspective  of  the  corner  B  by  a  horizontal  and  a  perpendicu- 
lar, of  B'  by  a  horizontal  and  a  vertical,  of  C'  by  a  diagonal  and 
a  perpendicular,  of  C  by  a  vertical  and  a  perpendicular,  and  of  E' 
by  a  perpendicular  and  a  diagonal.  The  perspective  of  E  may  be 
found  by  the  perspective  of  the  horizontal  ce,  and  that  of  the 
perpendicular  a  e,  but  a  better  way  would  be  to  determine  it  by 
the  perspectives  of  the  vertical  e'  e  and  the  horizontal  c  e,  be- 
cause the  intersection  of  the  last  two  lines  meeting  more  nearly  at 
right  angles  gives  a  more  accurate  result.  Avoid  determining  the 
perspectives  of  points  by  lines  which  make  a  small  angle  at  their 
intersection. 

To  find  the  perspective  of  the  shadow  of  the  cube:  The  perspective 
of  the  line  of  shade  is  V  b  c  e  e!\  this  line  casts  the  perspective  of 
the  outline  of  the  shadow.  The  perspective  of  the  shadow  of  b' 
is  itself,  since  it  is  in  the  plane  of  the  shadow.  To  determine  the 
perspective  of  the  shadow  of  b,  draw  b'  R*,  the  perspective  of  the 
projection  on  the  T  plane  of  the  ray  of  light  through  B,  and  b  R,,, 
the  perspective  of  the  ray  of  light  through  B;  where  these  two 
lines  meet  will  be  the  perspective  of  the  shadow  of  B  on  the  T 
plane.  The  line  b  c  is  the  perspective  of  a  perpendicular  which 
is  parallel  to  the  T  plane,  hence  its  shadow  on  the  T  plane  will 
be  a  perpendicular,  and  the  perspective  of  its  shadow  will  vanish 
at  S,,.  b,  S0  is  the  perspective  of  the  indefinite  shadow  of  b  c, 
and  where  the  line  c  R^  intersects  this  line  is  the  perspective  of 
the  shadow  of  the  corner  C  on  the  T  plane.  Ci  e\,  parallel  to 
c  e,  is  the  perspective  of  the  shadow  of  the  edge  C  E,  and  e\  tf  that 
of  the  edge  E  E'.  . 


PLATE  35. 


PERSPECTIVE 


D., 


100 


MECHANICAL   DRAWING. 


66. 


PLATE  36. 


Find  the  perspective  of  a  prism,  the  perspective  of  its  shade,  and 
the  perspective  of  its  shadow  on  the  horizontal  plane  (T) 
upon  which  it  rests.  The  height  is  3",  the  base  rectangular 
2"  by  1.5",  with  a  wider  face,  making  an  angle  of  30  degrees 
with  V  and  the  bases  parallel  to  H. 

Eh  is  at  (4.25",  6")  and  Ev  at  (4.25",  4"). 

Bv  IV,  the  projection  on  V  of  the  edge  B  B',  may  be  taken  as 
the  line  of  heights. 

The  trace  on  H  of  the  picture-plane  is  at  51=4.5", 

The  horizon  at  51=4.25", 

Sj,  at  #=6.25", 

Dv  at  #=12", 

R;,  at  x—  n",  and 

P,,  is  the  vanishing-point  of  the  perspectives  of  the  edge  B  C  and 
all  edges  parallel  to  it.  It  is  found  thus:  Produce  Eh  Q, 
until  it  meets  the  trace  of  the  picture-plane,  then  m  n  :  m  k= 
S»  T>v  :  S»  Ft,.  The  vanishing-point  of  the  perspectives 


of  the  edge  A  B  and  all  others  parallel  to  it  might  be  found 
in  a  similar  way,  but  since  the  point  falls  outside  the  limits 
of  the  drawing  it  is  not  used. 

Find  the  perspectives  a'  and  V  by  perpendiculars  and  diagonals, 
draw  a'  Pv  and  the  point  of  intersection  with  the  line  £„'  Sw, 
the  perspective  of  a  perpendicular  through  E',  will  be  the 
perspective  of  E'.  b  is  the  intersection  of  b'  b  and  B0  S«, 
the  perspectives  of  a  vertical  and  a  perpendicular  respectively 
through  B.  b  c  vanishes  at  P«.  a  is  the  intersection  of 
a'  a  and  h  Dr,  the  perspectives  of  a  vertical  and  a  diagonal 
respectively  through  A. 

The  only  face  in  view  which  is  in  the  shade  is  a'  a  e  e'. 

The  perspective  of  the  shadow  of  the  edge  A  A'  on  the  T  plane 
is  a'  ai,  vanishing  at  R&.  a\  e\,  the  perspective  of  the  shadow 
of  A  E,  vanishes  at  Pv.  c'  c±  vanishes  at  R&,  and  the  point 
where  the  line  c  Rv  meets  the  line  c?  Rj,  is  the  perspective 
of  the  shadow  of  the  corner  C. 


PLATE  35. 


101 


102 


MECHANICAL   DRAWING. 


PLATE  37. 


67.  Find  the  perspective  of  the  frustum  of  a  cylin- 
der as  given  in  Plate  24  (c),  and  of  its  shadow  on  a  hori- 
zontal plane  (T)  upon  which  it  rests.  The  projection  on 
H  of  the  center  of  the  lower  base  is  at  (5",  6"),  and  the 
projection  on  V  of  the  same  point  at  (5",  1.5")- 

The  picture-plane  and  V  are  one  and  the  same  plane. 

The  trace  on  H  of  the  picture-plane  is  at  y  =  5". 

The  horizon  at  ^  =  4.25". 

SB  is  at  ^  =  7.25". 

D;  at  *  =  2.5", 

Rh  at  x  =  g",  and 

Ryaty  =  2.5". 

A^  B^  is  the  projection  on  V  of  the  upper  base. 

Find  a'  by  a  perpendicular  and  a  diagonal  through  A, 
and  a  by  a  vertical  and  a  perpendicular  through  A. 

Find  the  perspective  of  other  points  in  the  boundary 
of  the  lower  base  by  perpendiculars  and  diagonals, 
and  the  perspective  of  corresponding  points  immediately 
above  by  verticals  and  perpendiculars.  The  perspective 
of  each  perpendicular  has  a  point  on  the  line  Av  B,,, 
which  is  the  projection  of  the  point  on  V,  and  it  vanishes 
at  Su.  Find  a  sufficient  number  of  points  in  the  bases 
to  accurately  outline  their  perspectives  by  joining  them 
with  an  irregular  curve. 


To  find  the  perspective  of  the  shadow  on  the  T  plane 
draw  lines  through  Rh  tangent  to  the  lower  base  at  d' 
and  e',  and  through  RB  tangent  to  the  upper  base  at  e 
and  d.  d  d'  and  e  e'  are  the  perspectives  of  the  lines  of 
shade  on  the  surface  of  the  cylinder,  and  e'  Rh  and  d'  Rk 
are  the  indefinite  perspectives  of  these  lines  of  shade  on 
the  T  plane  respectively,  and  where  e  Rv  meets  e'  Rh  is 
the  limit  on  one  side  and  similarly  d^  is  the  limit  on  the 
other.  Through  the  perspective  of  points  on  the  lower 
base  draw  lines  to  Rh,  and  from  the  corresponding  points 
on  the  upper  base  draw  lines  to  R^;  where  these  lines 
meet  will  be  the  perspectives  respectively  of  the 
shadows  of  the  points  on  the  T  plane.  For  example, 
k:  is  the  perspective  of  the  shadow  of  k,  k'  Rh  meet- 
ing k  Rv  at  ki.  In  this  way  the  line  Ci  kt  d:  is  deter- 
mined. 

Without  discussing  the  reasons  for  the  effect  of 
light  on  a  curved  surface  as  seen  by  the  eye,  it  is 
sufficient  to  say  in  this  connection  that  a  surface  in 
the  shade  will  appear  to  grow  lighter,  and  con- 
versely a  surface  in  the  light  will  appear  to  grow 
dearkr  as  the  surfaces  recede.  In  shading  the  perspec- 
tive of  the  surface  of  the  cylinder  students  should 
observe  these  rules. 


PLATE  37. 


103 


104 


MECHANICAL  DRAWING. 


PLATE  38. 


68.  Find  the  perspective  of  the  cube  with  a  section 
cut  from  one  face  and  having  a  square  hole  running 
through  it,  as  shown  in  Plate  13  (b). 

Bk  is  at  (3.25",  5-75")  and  B.  at  (3.25",  2$")- 

The  plane  of  the  section  is  taken  parallel  to  the  picture- 
plane. 

The  trace  of  the  picture-plane  on  H  is  at  y  =  5". 

The  horizon  is  at  y  =  4.25". 

Sv  is  at  *  =  6.25", 

D,,  at  x  =  12", 

R;,  at  x  =  10",  and 

Reaty=o.5". 

DB  and  D/  are  at  equal  distances  from  S,,. 

P0  is  the  vanishing-point  of  the  perspective  of  the 
edge  B  N,  and  all  edges  parallel  to  it,  found  as  explained 
in  Art.  66. 

The  horizontal  lines  o,  I,  2,  etc.,  are  at  heights  above 
the  T  plane  equal  to  the  heights  of  the  edges  E  H,  D  M, 
the  point  C,  etc.,  respectively. 

In  finding  the  perspectives  of  the  different  corners  of 
the  cube  and  the  hole,  the  following  method  is  observed: 
Find  the  perspective  of  a  point  on  the  T  plane,  directly 
under  the  corner,  by  a  perpendicular  and  a  diagonal. 
This  is  the  perspective  of  the  projection  of  the  corner  on 
the  T  plane.  A  vertical  line  is  drawn  through  this 
point  and  is  the  perspective  of  a  vertical  through  the 
corner.  The  point  where  the  perspective  of  a  perpen- 
dicular, a  horizontal,  or  a  diagonal  through  the  corner 
meets  this  vertical  line  is  the  perspective  of  the  corner. 


•  This  method  serves  to  aid  in  finding  the  perspective  of 
the  shadow,  for  the  perspective  of  the  projection  of  the 
corner  on  the  T  plane  is  one  point  of  the  perspective 
of  the  projection  of  the  ray  of  light  which  passes 
through  the  corner. 

b"  is  the  perspective  of  the  projection  of  B  on  the  T 
plane,  b"  b  the  perspective  of  the  vertical  through  B, 
and  r  b,  vanishing  at  S  B,  the  perspective  of  the  perpen- 
dicular through  B,  giving  b  the  perspective  of  the  corner 
B.  c"  is  the  perspective  of  the  projection  of  the  corners 
C  and  C'  on  the  T  plane,  c"  c  the  perspective  of  the  ver- 
tical, and  s  c  and  s'  c'  the  perspectives  of  the  perpendicu 
lars  through  C  and  C'  respectively.  The  other  corners 
are  found  in  a  similar  manner. 

The  perspective  of  the  line  of  shade  is  e'  g  k  h  n  b  c  e', 
and  the  perspective  of  its  shadow  on  the  T  plane 
e'  gi  ki  hi  HI  bi  c\  e' . 

To  find  the  perspective  of  the  shadow  of  the  corner  g, 
for  example,  draw  from  g",  the  perspective  of  the 
projection  of  the  corner  G,  a  line  to  RA,  the  vanish- 
ing-point of  the  projection  of  rays  of  light  on  the 
T  plane,  and  a  second  line  from  g  to  Rv;  the  point 
where  these  lines  intersect,  is  the  perspective  of  the 
shadow  of  g  on  the  T  plane.  Again,  draw  k"  Rh 
and  k  Rv,  and  the  point  where  these  lines  intersect 
gives  k^  Similarly  all  the  other  points  of  the  outline 
of  the  perspective  of  the  shadow  may  be  found.  The 
face  e'  g  k  h'  is  the  only  face  in  the  shade  which  is  in 
view. 


PLATE  38. 


105 


106 


MECHANICAL   DRAWING. 


PLATE  39. 

(Continuation  of  the  shadow  of  Plate  38.) 


Find  the  perspective  of  the  shadow  of  the  edge  of  the 
hole,  d  e",  on  the  face  of  the  hole  d  m  w"s,  and  deter- 
mine the  outline  of  the  bright  spot  in  the  shadow  on 
the  T  plane  caused  by  the  light  shining  through  the  hole. 

The  face  of  the  hole  d  e"  h"  m  is  in  the  shade,  this 
being  the  only  face  in  view  that  is  in  the  shade. 

Find  the  perspectives  of  the  shadows  on  the  T  plane 
of  all  the  edges  of  the  hole,  and  the  outline  of  the  bright 
spot  will  thus  be  determined.  If,  however,  it"  is  clear 
to  the  student  which  edges  cast  the  shadows  which 
are  the  outlines  of  the  bright  spot,  it  will  be  necessary  to 


find  the  perspectives  of  the  shadows  of  those  edges  only. 

Find  the  perspectives  of  the  shadows  m  w" ,  w"  t, 
e"  /,  and  d  e"  on  the  T  plane  in  the  usual  way.  These 
are  the  edges,  parts  of  which  will  cast  the  shadows  which 
are  the  outlines  of  the  bright  spot  in  the  shadow  on  the 
T  plane. 

At  Zi,  where  d^  e"  meets  m\  ze>i,  draw  a  line  to  R0  and 
extend  it  backwards  to  o.  This  is  the  perspective  of  the 
ray  of  light  which  limits  the  shadow  of  d  e"  on  the  T 
plane,  o  e"  casts  the  shadow  z±  e^"  on  the  T  plane, 
and  d  o  the  shadow  d  z  on  the  face  d  m  w"s. 


PLATE  39. 


M 


D; 


107 


PART  II. 

PROBLEMS  IN  DESCRIPTIVE  GEOMETRY. 


GENERAL  INSTRUCTIONS  AND  CONVENTIONS. 


I.  Instructions. — The  planes  of  projection  and  auxiliary 
planes  are  assumed  to  be  transparent,  and  given  and 
required  planes  opaque. 

If  a  given  line  is  in  view,  its  projections  are  to  be  drawn 
medium,  full,  black;  and  if  hidden,  light,  broken,  black. 

If  a  required  line  is  in  view,  its  projections  are  to  be 
drawn  heavy,  full,  black,  and  if  hidden,  light,  broken,  black. 

If  the  trace  of  a  given  plane  is  in  view,  it  is  to  be  drawn 
medium,  full,  black;  and  if  hidden,  light,  broken, 
black. 

Construction  lines  are  to  be  drawn  light,  full,  red; 
lines  joining  two  projections  of  the  same  point,  light,  dotted, 
red;  and  dimension-lines,  when  used,  are  light,  full,  red, 
but  the  arrow-points  marking  the  limits  and  the  nu- 
merals are  black. 

Traces  of  auxiliary  planes  (planes  used  for  construction ) 
are  to  be  light,  dash-and-two-dot,  red. 

Given  points  are  to  be  enclosed  by  very  small  circles  in 
black. 

The  foregoing  rules  do  not  apply  to  the  projections  of 
the  outlines  of  surfaces. 

For  the  size  of  the  plates  and  their  subdivision,  and 
for  explanation  of  coordinate  axes  and  ordinates,  see 
Part  I,  Art.  19. 

2.  Conventions. — The  four  dihedral  angles  formed  by 


the  intersection  of  the  H  and  V  planes  will  be  designated 
as  follows: 

The  first  angle,  above  H  and  in  front  of  V; 

The  second  angle,  above  H  and  behind  V; 

The  third  angle,  below  H  and  behind  V;  and 

The  fourth  angle,  below  H  and  in  front  of  V. 

The  capitals  H,  V,  P,  and  S  will  designate  the  hori- 
zontal, vertical,  profile,  and  supplementary  planes  of  pro- 
jection, respectively. 

An  auxiliary  plane  will  be  designated  by  either  R,  T, 
or  U. 

The  ground-line  will  be  called  H  V  between  the  hori- 
zontal and  vertical  planes,  between  the  horizontal  and 
profile  planes  H  P,  and  between  the  vertical  and  profile 
planes  V  P. 

A  point  in  space  will  be  designated  by  a  capital  letter, 
e.g.  P,  and  its  projections  on  H,  V,  P,  and  S,  by  Ph, 
Pv,  Pp,  and  Ps,  respectively. 

A  line  in  space  will  be  designated  by  referring  to  any 
two  points  in  the  line,  as  the  line  M  N,  and  its  projections 
to  the  corresponding  projections  of  those  points. 

The  problems  from  i  to  18  are  to  be  constructed  and 
numbered  consecutively  in  the  squares  of  their  respective 
plates  (see  Part  I,  Diagram  A).  The  ground-line,  H  V, 
will  bisect  the  square  in  each  of  the  problems. 

Ill 


112 


MECHANICAL  DRAWING. 


Problems. 
PLATE  I. 


Prob.  i.  Represent  four  points  A,  B,  C,  and  D: 

A  in  the  first  angle  ift"  from  H,  ft"  from  V,  x=W,  * 

B  in  the  second  angle  i&"  from  H,  &"  from  V,  x=  if"; 

C  in  the  third  angle  i&"  from  H,  ft"  from  V,  *=2j"; 

D  in  the  fourth  angle  i  A"  from  H,  W'  from  V,  x=  2ft". 
Prob.  2.  Represent  two  right  lines  MN  and  OP: 

MN  in  the  third  angle,  perpendicular  to  H,  i"  from  V, 
if"  long; 

M  lying  in  H,.Y=H"; 

OP  in  the  third  angle  parallel  to  V  and  H"  from  V; 

O,  f'fromH,  *=ii"; 

P,  iJ"fromH,  x=3". 
Prob.  3.  Represent  the  plane  of  a  line  MN  and  a  point  P: 

(#  represents  the  distance  of  the  projections  of  the  point  from 
the  vertical  axis  of  the  square.} 


MN  in  the  third  angle,  parallel  to  AB,  ij"  from  H, 

I"  from  V; 

P  in  the  third  angle,  i"  from  H,  j"  from  V,  x=  rj". 
Prob.  4.  Represent  a  plane,  R,  in  the  third  angle  passing  through 

three  points  A,  B,  and  C: 

A  in  the  first  angle,  i"  from  H,  ij"  from  V,  *=i"; 
B  in  the  third  angle,  J"  from  H,  \"  from  V,  x=  ii"; 
C  in  the  fourth  angle,  ij"  from  H,  f"  from  V,  #=3". 
Prob.  5.  Represent  a  plane  R  in  the  third  angle  passing  through 

a  point  P  and  perpendicular  to  a  line  MN: 
P  in  the  third  angle,  i"  from  H,  J"  from  V,  x=2\"\ 
M  in  the  third  angle,  if  from  H,  J"  from  V,  *=$"; 
N  in  the  second  angle,  J"  from  H,  i"  from  V,  x=  ij". 
Prob.  6.  Given  the  points  and  plane  of  Problem  5,  to  represent 

the  point  where  MN  pierces  R  at  G,  and  also  show  the 

true  length  of  the  line  joining  P  with  G. 


PROBLEMS   IN   DESCRIPTIVE    GEOMETRY. 


113 


PLATE  II. 


Prob.    7.  Represent  in  the  third  angle  the  intersection  of  two 

planes,  R  and  T. 

The  traces  of  the  R  plane  meet  at  x=y. 
The  traces  of  the  T  plane  meet  at  x=  2 \". 
The  H  trace  of  the  R  plane  makes  with  HV  an  angle 

of  75°- 
The  V  trace  of  the  R  plane  makes  with  HV  an  angle 

of  60°. 
The  H  trace  of  the  T  plane  makes  with  HV  an  angle 

of  120°. 
The  V  trace  of  the  T  plane  makes  with  HV  an  angle 

of  105°. 
Prob.    8.  Show  the  true  size  of  the  angle  represented  between  the 

lines  MK  and  OK. 
MK  is  in  the  third  angle  parallel  to  HV,  i"  from  H, 

i"  from  V. 

O  K  is  in  the  third  angle. 
O  is  ij"  from  H,  ij"  from  V,  x=z\". 
ForK,  *=ii". 
Prob.    9.  Represent  on  a  profile  plane,  through  the  point  O,  the 

solution  of  Prob.  8,  looking  in  the  direction  from 

right  to  left. 


For  the  V  P  trace,  x=  f". 
Prob.  10.  Represent  the  intersection,  MN,  of  two  planes  parallel 

to  HV  lying  across  the  third  angle. 
The  T  plane  makes  an  angle  of  30°  with  H  and  60° 

with  V;  the  H  trace  is  i"  from  HV. 
The  R  plane  makes  an  angle  of  75°  with  H  and  15°  with 

V;  the  H  trace  is  f"  from  HV. 
Prob.  ii.  Represent  the  point  where  a  line,  M  N,  pierces  a  plane, 

T,  and  the  projection  of  the  line  on  the  plane. 
MN  is  in  the  third  angle,  parallel  to  HV,  i"  from  H, 

i"  from  V. 
The  H  trace  of  the  plane  makes  an  angle  of  30°  with 

HV,  above. 
The  V  trace  of  the  plane  makes  an  angle  of  60°  with 

HV,  below. 

The  traces  intersect  at  x=  i". 
Prob.  12.  Represent  the  point  where  a  line,  MN,  pierces  a  plane, 

T. 

MN  is  in  the  third  angle. 
M  is  i"  from  H,  J"  from  V,  x=  i  J". 
N  is  \"  from  H,  i"  from  V,  x=  i  J". 
The  plane  is  taken  as  in  Prob.  ii. 


114 


MECHANICAL   DRAWING. 


PLATE  III. 


Prob.  13.  Trisect  the  angle  between  two  right  lines,  MN  and  MP, 

and  show  the  projections  of  the  trisecting  lines. 
MN  is  in  the  third  angle,  parallel  to  H,  \"  from  H. 
Mis  i  J"  from  V,  x=f". 
Nisi"  from  V,  *=2j". 

P  is  in  the  third  angle,  i"  from  H,  f"  from  V,  x=  2". 
(Suggestion:  Revolve  the  plane  of  the  two  lines  about 

MN,  as  an  axis,  until  it  is  parallel  to  H. 
Prob.  14.  Represent  a  plane  passing  through  a  point    Q    and 

parallel  to  the  lines  MN  and  OP. 
Q  is  in  the  third  angle,  i"  from  H,  |"  from  V,  x=$". 
MN  is  in  the  third  angle,  parallel  to  HV,  \"  from  H, 

i"  from  V. 

O  is  in  the  third  angle,  J"  from  H,  J"  from  V,  x=  i". 
P  is  in  the  third  angle,  ij"  from  H,  J"  from  V,  x=2\". 
Prob.  15.  Find  the  true  distance  from  a  point  P  to  a  line  MN. 


P  is  in  the  third  angle,  i"  from  H,  \"  from  V,  x=  i". 
M  is  in  the  third  angle,  i"  from  H,  ij"  from  V,  x=  J". 
N  is  in  the  third  angle,  i"  from  H,  J"  from  V,  x=2\". 
Prob.  16.  Find   the  common  perpendicular,   X  Y,  between  the 

lines  MN  and  OP. 
MN  is  in  the  third  angle,  perpendicular  to  H,  i"  from 

V,  x=l". 

O  is  in  the  third  angle,  i"  from  H,  \"  from  V,  x=  ij". 

P  is  in  the  third  angle,  J"  from  H,  i J"  from  V,  x=  zj". 

Prob.  17.  Represent  an  oblique  plane,  T,  not  parallel  to  HV, 

making  an  angle  of  60°  with  H  and  45°  with  V,  and 

passing  through  a  point  P. 

P  is  in  the  third  angle,  i"  from  H,  £"  from  V,  #=3". 
Prob.  18.  Represent  a  line  making  an  angle  of  30°  with  H  and 

45°  with  V,  and  passing  through  a  point  P. 
P  is  in  the  third  angle,  i"  from  H,  J"  from  V,  x=  2$". 


GENERAL   INSTRUCTIONS    AND    CONVENTIONS. 


115 


6.  Shade-lines. —  The  projection  of  surfaces  in  the  prob- 
lems to  follow  will  not  be  sectioned-lined  to  represent 
shaded  parts,  but  shade-lines  will  indicate  which  of  the 
surfaces  in  view  are  in  the  shade.  Shade-lines  are  those 
representing  edges  which  join  surfaces,  or  faces,  one 
in  the  light  and  the  other  in  the  shade. 

In  curved  surfaces,  such  as  the  cylinder  and  the  cone, 
the  elements  of  shade  do  not  coincide  with  the  elements 
of  outline;  the  side  outlines  will  have,  therefore,  no  dis- 
tinction made  in  their  grade.  The  outline  of  the  bases 
should  conform  to  the  rule  which  will  be  given  later. 

The  H  and  V  projections  of  rays  of  light  will  each 
make  an  angle  of  45  degrees  with  H  V,  because  the  rays 
of  light  are  assumed  to  approach  the  H  and  V  planes 
from  the  first  angle,  in  the  direction  of  the  body  diagonal 
of  a  cube  whose  faces  are  parallel  to  H  and  V. 

When  a  body  is  made  up  of  plane  surfaces,  or  faces, 
in  order  to  ascertain  whether  a  face  is  in  the  light  or  in 
the  shade,  it  is  necessary  to  determine  whether  rays  of 
light  pierce  the  surface  uninterruptedly,  or  are  inter- 
cepted by  another  face.  For  example,  in  Fig.  8,  Part  I, 
to  determine  whether  the  face  A  B  E  F  is  in  the  shade, 
draw  the  trace  on  H  of  a  plane  of  rays,  R;,,  perpendicular 
to  H,  intersecting  the  faces  A'  B'  E  F  and  A  B  E  F  in 
the  lines  M  N  and  N  H  respectively.  If  any  ray  of 
light  in  this  plane  can  be  drawn  which  will  pierce  the 
face  A  B  E  F  before  it  pierces  the  face  A'  B'  E  F,  then 
the  face  A  B  E  F  is  not  in  the  shade ;  otherwise  it  is  in 


the  shade  because  the  other  face  intercepts  the  rays 
of  light.  The  ray  R  drawn  through  any  point  of  the 
line  M  N,  as  G,  in  the  face  A'  B'  E  F,  pierces  the  other 
face  at  H,  and  all  other  rays  of  light  piercing  the  face 
A'  B'  E  F  between  G  and  N  will  also  pierce  the  face 
A  B  E  F  between  N  and  H.  Therefore  the  face  A  B  E  F 
is  in  the  shade,  because  the  face  A'  B'  E  F  intercepts  the 
rays  of  light  in  their  path  towards  the  first-named  face. 

Again,  the  plane  of  rays  of  which  RA  is  the  trace  on 
H  cuts  from  the  face  A'  B'  C'  D'  the  line  M  P,  and  from 
the  face  C'  D'  K  L  the  line  P  Q.  The  ray  R',  lying  in 
this  plane  and  meeting  the  first-named  surface  in  W, 
does  not  intersect  the  line  P  Q,  which  also  lies  in  the 
same  plane  of  rays  and  in  the  face  C'  D'  K  L.  No  ray 
in  that  plane  drawn  through  any  point  in  the  line  M  P 
will  intersect  the  line  P  Q.  The  face  C'  D'  K  L  is  not  in 
the  shade  by  reason  of  the  interception  of  rays  by  the 
face  A'  B'  C'  D'. 

Having  determined  which  faces  of  a  body  are  in  the 
light  and  which  are  in  the  shade,  the  following  rule  is  to 
be  observed:  Those  lines  of  a  drawing  representing  edges 
between  two  faces,  both  in  the  light,  and  those  representing 
edges  between  two  faces,  both  in  the  shade,  are  to  be  drawn 
light  or  medium;  and  lines  representing  edges  between  two 
faces,  one  in  the  light  and  the  other  in  the  shade,  are  to  be 
drawn  heavier.  However,  lines  which  represent  hidden 
edges  are  to  be  drawn  light,  broken,  whatever  the  conditions 
of  light  and  shade. 


116 


MECHANICAL    DRAWING. 


In  profile  and  supplementary  projections  it  will  be 
assumed  that  the  object  represented  remains  stationary, 
while  the  planes  of  projection  change  their  positions; 
hence  the  shade-lines  are  the  same  lines  in  these  projections 
as  in  the  projections  on  H  and  V. 

The  direction  of  the  rays  of  light  as  given  above  is 
that  usually  adopted,  but  there  is  no  good  reason  other 
than  this  why  the  direction  may  not  be  assumed  at  the 


will  of  the  draftsman.  It  is  the  custom  of  some  drafts- 
men to  disregard  any  reference  to  light  and  shaded  faces, 
but  to  arrange  the  light  and  heavy  lines  to  produce  a 
pleasing  effect  in  the  drawing.  The  student  is  recom- 
mended to  observe  exact  methods  until  experience 
teaches  him  to  readily  assume  conditions  of  light  and 
shade  which  do  not  produce  glaring  inconsisten- 
cies. 


7- 


PLATE  IV. 


Prob.  19.  Represent  a  plane  tangent  to  a  prolate  spheroid 
in  the  third  angle,  at  a  point,  P,on  its  surface. 
The  transverse  axis  is  3"  long  perpendicular  to 
H,  if"  from  V;  x  =  &";  the  highest  point,  M, 
is  \"  from  H. 
The  conj  ugate  axis  is  2 \"  long. 


8. 


PLATE  V. 


Prob.  20.  Find  the  intersection  of  a  right  cone  with  a 

plane. 
Show  the  section  in  its  true  size,  make  a  profile 

of  the  lower  part  and  develop  it. 
The  cone  is  in  the  third  angle;  the  axis  vertical, 

3"  long,  2"  from  V  at  x  =  6^";  the  vertex  is 

in  H,  and  the  diameter  of  the  base  is  3". 
The  plane  is  perpendicular  to  V  and  cuts  the 

axis  at  a  point  ij"  from  the  vertex;    the  V 


The  point  P  is  on  the  upper  surface;   for  its  H 

projection  x  =  6",  y  =  6|". 
Note:   Take  a  whole  plate  for  this  problem  and 

those  to  follow  and  let  the  origin  be  at  the  lower 

left  corner  of  the  border-line  of  the  plate,  and 

let  H  V  divide  the  plate  in  halves. 


trace  of  the  plane  inclines  downwards  to  the 

right,  making  an  angle  of  45  degrees  with  HV. 
Let  the  axis  of  the  cone  in  profile  be  at  x  =  10". 
Let  the  transverse  axis  of  the  true  size  of  the 

section  be  parallel  to  H  V  at  #  =  io",  y  =  6J" 

for  the  middle  point. 
In  the  development,  or  pattern,  let  the  longest 

element  coincide  with  a  horizontal  line  at 
=   i"  and  the  vertex  be  at  #  =   ",  y  =  4£". 


GENERAL  INSTRUCTIONS    AND    CONVENTIONS. 


117 


PLATE  VI. 


Prob.  21.  The  same  as  Prob.  20,  except  that  the  cutting 
plane  is  to  be  taken  so  as  to  cut  a  section 


from   the  right  cone  which  shall  be  a  par- 
abola. 


10. 


PLATE  VII. 


Prob.  22.  The  same  as  Prob.  20,  except  that  the  cutting 
plane  shall  cut  a  section  which  shall  be  an 


hyperbola,  the  V  trace  making  an  angle  of 
75  degrees  with  H  V. 


ii. 


PLATE  VIII. 


Draw  an  oblique  cylinder,  the  elements  making  an 
angle  of  30  degrees  with  H  and  45  degrees  with  V,  in- 
clining downward  to  the  right  and  toward  V.     The  base  - 
in  H  is  a  circle  of  i"  radius. 

Find  the  curve  of  intersection  of  this  cylinder  with  V, 
project  the  cylinder  on  a  plane  parallel  to  the  elements 
to  show  their  true  lengths,  and  develop  the  cylin- 
der. 

TakeHV  at  y  =  3-75". 

For  the  center  of  the  base  in  H,  #  =  2.5",  y  =  6". 

Let  the  cylinder  lie  across  the  third  angle. 

To  find  the  true  lengths  of  the  elements  between  the 
bases  in  H  and  V,  draw  an  auxiliary  projection  of  the 


cylinder  on  a  plane  perpendicular  to  H  and  parallel  to 
the  elements.  Take  the  H  trace  of  this  plane  crossing 
H  Vat  x  =  6". 

Pass  a  right-section  plane  through  the  cylinder  at  a 
point  along  the  axis  i"  from  H;  show  one-half  the  section 
on  the  auxiliary  projection  by  revolving  it  on  a  line  which 
is  parallel  to  the  auxiliary  plane  lying  in  the  section  plane 
and  passing  through  the  axis. 

For  the  development,  let  the  right-section  line  be  rec- 
tified on  a  line  perpendicular  to  H  V,  at  x  =  io". 

Let  the  element  which  is  the  shortest  between  H  and 
the  right-section  plane  be  laid  out  on  a  horizontal  line 
through  the  middle  of  the  plate. 


118 


MECHANICAL   DRAWING. 


12. 


PLATE  IX. 


Prob.  23.  To  develop  an  oblique  cone  which,  is  repre- 
sented in  the  third  angle.  Find  the  inter- 
section of  the  cone  with  an  hemisphere  whose 
center  is  at  the  vertex,  and  whose  radius. is 
2." ' .  Develop  the  horizontal  projecting  cylin- 
der of  this  intersection,  and  use  this  cylinder 
in  developing  the  cone. 

The  base  of  the  cone  is  a  circle  whose  plane  is 
in  H,  the  center  at  x=5%,  y  =  6j,  and  the 
radius  ij". 


The  vertex  is  2|"  from  H  in  V,  at  #  =  4". 

Short  arcs  should  represent  the  hemisphere  so  as 
not  to  interfere  with  other  lines  of  the  drawing. 

In  the  development  of  the  projecting  cylinder 
of  intersection,  take  the  base  of  the  cylinder 
perpendicular  to  H  V  at  x  =  \",  and  let  the 
longest  element  coincide  with  H  V. 

In  the  development  of  the  cone,  take  the  vertex 
at  x  =  12",  y=4J",  and  the  shortest  element 
inH  V. 


13. 


PLATE  X. 


Prob.  24.  Find  the  intersection  of  three  cylinders,  A,  B, 

and  C,  whose  right-sections  are  circles,  and 

develop  each  cylinder. 
All  the  cylinders  are  in  the  third  angle. 
The  axis  of  A  is  vertical,  3"  long  and  is  2"  from 

V,  at  #  =  6.25".     The  upper  base  is  in  H,  and 

its  diameter  equals  3". 
Cylinder  B  is  to  the  left  of  A  and  its  axis  is 

parallel  to  HV,  1.5"  from  H  and  2.5"  from  V. 

The  plane  of  the  base  is  perpendicular  to 

H  V,  at  #  =  3.75",  and  the  diameter  of  the 

base  is  2". 


Cylinder  C  is  to  the  right  of  A;  its  axis  parallel 
to  V  and  inclined  30  degrees  to  H  upwards  to 
the  right.  The  axis  of  C  intersects  the  axis  of 
A  at  2.5"  from  H.  The  plane  of  the  base  is 
perpendicular  to  the  axis  with  its  center  at 
x  =  8.75".  The  diameter  of  the  base  equals  2". 

To  develop  B  let  the  base  rectify  on  a  line  per- 
pendicular to  H  V,  at  x  =  0.75";  the  longest 
element  of  the  cylinder  coinciding  with  HV. 

To  develop  C  let  the  base  rectify  on  a  line  per- 
pendicular to  H  V,  at  #  =  11.75";  the  lowest 
element  coinciding  with  H  V. 


GENERAL   INSTRUCTIONS   AND    CONVENTIONS. 


119 


14. 


PLATE  XI. 


Prob.  25.  To  develop  A  of  Plate  X,  let  the  base  be  recti- 
fied on  a  line  parallel  to  HV  and  1.5"  below 


it;   the    element   nearest    V   to   be   at   x •- 
6-75". 


IS- 


PLATE  XII. 


Prob.  26.  Find  the  intersection  of  a  regular,  hexagonal 
prism,  A,  and  a  right  cylinder,  B,  with 
another  regular,  hexagonal  prism,  C,  and 
develop  each  surface. 

Prism  C  has  its  axis  perpendicular  to  H  and 
parallel  to  V,  two  of  its  sides  perpendicular  to 
V,  and  is  3"  high.  The  side  of  the  hexag- 
onal base  is  1.5". 

In  the  projection  on  H,  the  center  of  the  base 
is  at  #  =  6.25",  y  =  6.25". 

The  projection  on  V  of  the  upper  base  is  at 

y=4-25". 

Prism  A  is  parallel  to  H  and  makes  an  angle  of 
30  degrees  with  V,  on  the  left  of  C,  its  axis 
intersecting  the  axis  of  C.  Two  of  the  sides 
of  the  prism  are  parallel  to  H.  The  side  of 
the  hexagonal  base  is  i".  The  plane  of 
the  base  is  perpendicular  to  the  axis  of  the 


prism,  the  center  of  the  H  projection  of  the 
base  being  at  #  =  3.75".  The  V  projection  of 
the  axis  is  at  7  =  2.5". 

Cylinder  B  is  parallel  to  both  H  and  V,  on  the 
right  of  C,  the  H  projection  of  its  axis  being 
at  y  =  6",  and  the  V  projection  at  y  =  2.5". 
The  radius  of  the  base  =  i",  and  its  plane  is 
perpendicular  to  the  axis;  the  H  projection 
of  the  center  of  the  base  being  at  x  = 

8.75". 
To  develop  A,  rectify  the  hexagon  of  the  base 

on  a  line  perpendicular  to  H  V,  at  #  =  0.5"; 

the  middle  line  of  the  upper  face  coinciding 

with  the  middle  line  of  the  plate. 
To  develop  B,  rectify  the  circumference  of  its 

base   on   a   line   perpendicular   to   H  V,    at 

#  =  12",  the  longest  element  coinciding  with 

the  middle  line  of  the  plate. 


120 


MECHANICAL    DRAWING. 


16. 


PLATE  XIII. 


Prob.  27.  To  develop  C  of  the  previous  plate,  rectify  the 
hexagon  of  the  lower  base  on  a  line  parallel 


to  H  V,  at  7  =  2.5",  so  that  the  edge  nearest 
V  shall  be  at  x  =  6.25". 


17- 


PLATE  XIV. 


Prob.  28.  Find  the  intersection  of  a  right  cylinder,  A,  and 
a  regular,  hexagonal  prism,  B,  with  a  regular, 
hexagonal  pyramid,  C,  and  develop  each 
surface. 

Pyramid  C  has  its  axis  perpendicular  to  H, 
3.5"  long,  the  H  projection  at  #  =  7.25", 
y  =  6.5".  One  side  of  the  base  is  parallel  to 
V.  The  sides  of  the  base  are  each  1.5"  long. 
The  apex  of  the  pyramid  is  at  4.75"  in  the 
V  projection. 

Cylinder  A  is  to  the  left  of  C,  has  its  axis  parallel 
to  both  H  and  V,  and  its  axis  intersects  the 
axis  of  the  pyramid.  The  V  projection  of 
the  axis  is  at  y  =  2.25".  The  plane  of  the 
base  is  perpendicular  to  both  H  and  V  at 
x  =  5.25".  The  radius  of  the  base  =  0.75". 


Prism  B  is  in  front  of  C,  and  has  its  axis  parallel 
to  H  and  perpendicular  to  V,  the  axis  inter- 
secting the  axis  of  the  pyramid.  The  V  pro- 
jection of  the  axis  is  at  y  =  3-5".  The  sides 
of  the  base  are  each  0.5"  long.  The  plane  of 
the  base  is  parallel  to  V,  and  its  H  projection 
is  at  y  =  5.5". 

Draw  a  profile  projection  of  the  three  surfaces, 
the  projection  of  the  axis  of  C  being  at 

*  =  2.75". 

Develop  A,  the  base  rectifying  on  a  line  per- 
pendicular to  H  V,  at  #  =  n",  the  highest 
element  coinciding  with  a  line  at  y  =  4-25". 

Develop  B,  the  base  rectifying  on  a  line  perpen- 
dicular to  H  V,  at  x  =  1.25",  the  middle  of  the 
upper  face  coinciding  with  a  line  at  y=6.5". 


PROBLEMS   IN    DESCRIPTIVE    GEOMETRY. 


121 


18. 


PLATE  XV. 


Prob.  29.  Develop  C  by  taking  the  apex  at  #  =  6.25", 
y  =  6",   and  making   the   edge   to   the   left 


19. 


PLATE  XVI. 


Prob.  30.  Draw  the  isometric  projection  (Art.  54,  Part 
I)  of  the  drawing- table  whose  dimensions 
are  given  on  the  accompanying  sketch. 


parallel  to  V  coincide  with  a  line  perpendicu- 
lar to  H  V,  at  ^=6.25". 


The  bottom  of  the  drawer  is  raised  \"  from 
the  bottom  of  the  sides.  The  front  and  side, 
pieces  are  halved  together  at  their  joint. 


-1 

4 

lr-x                   3?                   X 

iM 

I 

( 

J 

J 

I 

-      ai/*    v 

5 

E 

o                •  ' 
x         $         x 
i 

1 

* 

K' 

.T 

T 

•i 

1 

—/  I-IM-                  ^- 

r 

\ 

FIG.  i. 


Show  the  second  drawer  open  and  projecting 
8".  The  thickness  of  the  front  and  sides  of 
the  drawer  is  i",  that  of  the  bottom  £". 


Let  the  scale  be  i"  to  i'. 

Draw  the  dimension-lines  and  insert  lengths. 

Letter  the   title  "Drafting-table,"   above  the 


122 


MECHANICAL   DRAWING. 


drawing,  and  make  below  it  a  scale  figured 
and  lettered. 
After    the    drawing    is    completed,    draw    the 


border-line  of  the  usual  dimensions,  sym- 
metrically placed  with  relation  to  the  draw- 
ing, and  cut  the  plate  the  usual  size. 


20. 


PLATE  XVII. 


Prob.  31.  Draw  a  pseudo-perspective  of  the  same 
drawing-table,  with  dimensions  and  scale 
the  same.  The  border  is  to  be  placed 


symmetrically  around  the  drawing  as  in  the 
previous  problem,  but  not  till  it  is  com- 
pleted. 


21.  Shades  and  Shadows. — There  are  no  new  principles 
of  Descriptive  Geometry  involved  in  the  subject  of  shades 
and  shadows.  The  direction  of  the  parallel  rays  of  light 
may  be  that  of  the  body-diagonal  of  a  cube  (Art.  34, 
Part  I),  or  that  fulfilling  any  suitable  condition  sug- 
gested to  the  draftsman. 

Any  body  upon  which  the  light  shines  will  be  divided, 
by  a  distinct  line  or  lines,  into  a  light  part  and  a  shaded 
part  (Art.  34,  Part  I).  This  dividing  line  is  called  the 
line  of  shade,  and  is  the  line  of  tangency  of  a  surface,  or 
surfaces,  made  up  of  rays  of  light  tangent  to  the  body. 
That  portion  of  space  from  which  the  light  is  cut  off  by 
this  body  is  said  to  be  the  indefinite  shadow  of  the  body. 
If  any  other  body  is  placed  within  this  indefinite  shadow, 
a  definite  shadow  will  be  cast  upon  it  by  the  first 
body. 


To  find  the  line  of  shade  on  a  body  it  is  necessary  to 
determine  the  intersection  of  the  tangent  surface  of  rays 
with  the  body.  The  character  of  this  surface  of  rays  is 
determined  by  the  form  of  the  body,  and  may  be  made 
up  of  planes  or  cylindrical  surfaces,  or  both. 

To  find  the  shadow  of  one  body  on  another  it  is  neces- 
sary to  find  the  intersection  with  the  second  body  of  this 
surface  of  rays  tangent  to  the  first  body.  Since  the  tan- 
gent surface  of  rays  is  wholly  made  up  of  planes  or  cylin- 
drical surfaces,  one  or  both,  the  problem  reduces  it:elf 
to  finding  the  intersection  of  planes  and  cylindrical  sur- 
faces with  surfaces  of  all  other  kinds.  And  since  the 
tangent  surfaces  are  made  up  of  rays  of  light,  the  problem 
may  be  further  reduced  to  finding  where  right  lines  inter- 
sect surfaces  of  different  kinds.  (See  Plates  33  and  34, 
Part  I.) 


PROBLEMS   IN   DESCRIPTIVE    GEOMETRY. 


123 


22. 


PLATE  XVIII. 


Prob.  32.  Draw  the  projections  on  H  and  V  of  an  hex- . 
agonal  pillar,  resting  on  a  square  base  and 
surmounted  by  a  square  cap-block.  Find 
the  shade  on  the  pillar,  the  shadow  of  the 
cap  on  the  pillar,  of  the  pillar  on  the  base, 
and  the  shadow  of  the  whole  on  a  horizontal 
plane  coinciding  with  the  lower  face  of  the 
base.  The  projection  of  rays  of  light  on  H 
and  V  are  to  make  an  angle  of  45  degrees 
with  H  V.  The  object  is  to  be  assumed  in 
the  third  angle. 

The  cap  is  2"  square  and  J"  thick,  with  one 
edge  parallel  to  V. 


The  base  is  2"  square  and  £"  thick,  with  one 

edge  parallel  to  V. 
The  pillar  is  2"  high  between  the  cap  and  base, 

and  a  side  of  its  base  is  £•". 
The  planes  of  two  of  the  faces  of  the  pillar  are 

perpendicular  to  V. 
The  axial  line  of  the  pillar  is  projected  on  H  at 

(5">  4i")>  and  the  lower  face  of  the  base  is 

aty-f". 
(After  the  drawing  is  completed  make  a  tracing 

of  it  for  use  in  a  problem  to  follow  later. ) 
Note:  For  a  more  extended  course  in  shades 

and  shadows  the  problems  of  Part  I   may 

be  used. 


124 


MECH'ANICAL   DRAWING. 


23.  Perspective. — Perspective  differs  from  orthographic 
projections  in  that  the  projecting  lines  of  points  are  not 
•parallel,  but  diverge  from  some  near  point  which  is  the 
position  of  the  eye.     The  orthographic  projections  of  a 
point  on  H  and  V  are  required  in  order  to  determine  its 
position  in  space,  so  that  orthographic  projections  must 
be  used  to  find  perspectives. 

The  perspective  is  usually  made  upon  the  V  plane, 
which  is  then  called  the  picture-plane. 

The  orthographic  projections  of  the  position  of  the  eye 
determine  what  is  called  the  point  of  sight,  and  the 
orthographic  projections  of  the  line  projecting  a  point  on 
the  picture-plane  determine  what  is  called  a  visual  ray; 
and  the  point  where  this  visual  ray  pierces  V  (the 
picture-plane)  is  called  the  perspective  of  the  point  (Art. 
63,  Part  I). 

The  eye  (point  of  sight)  is  located  in  the  fourth  angle, 
and  the  object  to  be  drawn,  in  the  third  angle. 

Example:  In  Fig.  2,  S  is  the  point  of  sight,  S  P  a 
visual  ray,  and  p,  where  the  visual  ray  to  the  point  P 
pierces  the  picture-plane,  the  perspective  of  the  point  P. 

24.  Vanishing-point. — The   perspectives  of  all  parallel 
lines  will  have  a  common  point.     This  point  is  called  the 
vanishing-point   of   that   system   of  parallel  lines.     The 
vanishing-point  of  any  system  of  parallel  lines  may  be 
determined  by  finding  where  a  parallel  line  through  the 
point  of  sight  pierces  the  picture-plane.     Because  if  visual 
rays  are  drawn  to   two  different  points  of  any  one   of 


these  lines,  they  will  determine  a  plane,  called  a  visual 
plane,  the  intersection  of  which  with  the  picture-plane 
is  the  indefinite  perspective  of  the  line.  The  parallel  line 
and  the  line  through  the  point  of  sight  parallel  to  it, 
determine  the  same  plane;  hence  where  the  parallel 


7>4 


FIG.  2. 

line  through  the  point  of  sight  pierces  the  picture-plane 
is  a  point  in  the  indefinite  perspective  of  the  line.  This 
paralle'  line  through  the  point  of  sight  is  common  to  the 
visual  planes  of  each  one  of  the  system  of  parallel  lines 
whose  intersections  with  the  picture-plane  determine 
their  indefinite  perspectives.  Therefore  the  point  in  the 
picture-plane  where  the  line  through  the  point  of  sight, 
parallel  to  the  system  of  parallel  lines,  pierces  it,  is  a 
point  in  the  indefinite  perspective  of  each  line;  hence 
their  vanishing-point. 


PROBLEMS   IN   DESCRIPTIVE    GEOMETRY. 


125 


The  vanishing-point  of  all  lines  lying  in  horizontal 
planes  is  in  the  horizon  (Art.  63,  Part  I).  A  diagonal  line 
is  a  line  lying  in  a  horizontal  plane  making  an  angle  of 
45  degrees  with  the  picture-plane.  Its  vanishing-point 
is,  therefore,  in  the  horizon  at  a  distance  from  the  vertical 
projection  of  the  point  of  sight,  called  the  center  of  the 
picture,  equal  to  the  distance  of  the  point  of  sight  from 
the  picture-plane.  (The  student  should  review  Arts.  63 
and  64,  Part  I,  before  attempting  the  following  problems 
in  perspective.) 

25.  To  Find  the  Vanishing-point  of  any  System  of  Par- 
allel Lines. — Having  given  the  center  of  the  picture,  S,,,  the 
point  of  distance,  DB,  and  the  angles  a  and  ft,  that  any 

v  ft  I 


s, 


FIG.  3. 


FIG.  4. 


system  of  parallel  lines  makes  with  V  and  H,  respectively, 
to  find  ihe  vanishing-point  of  the  lines.  It  is  required 
to  find  its  distances  from  a  vertical  line  through  the 
center  of  the  picture,  and  from  the  horizon. 

Let  the  triangle  S  SB  V  (Fig.  3 )  lie  in  the  vertical  pro- 


jecting  plane  of  a  visual  ray,  S  V,  passing  through  the 
point  of  sight  and  parallel  to  any  system  of  parallel  lines; 
S  S,,  is  perpendicular  to  the  picture  plane  and  a  is  the 
angle  the  visual  ray  makes  with  V. 

S  S 
In  the  right  triangle  sin  a  =  5^,",  but  S  SB  =  Df)  St.,  the 

distance  from  the  center  of  the  picture  to  the  distance 
point;  therefore 

S  V  =  SB  DB  cosec  a (i) 

and  St)V  =  SB  S  cot  a  =  Sv  D^  cot  a.     ...     (2) 

Again,  let  the  triangle  S  hV  (Fig.  4)  lie  in  the  hori- 
zontal projecting  plane  of  the  same  visual  ray,  S  V; 
and  let  S  h  be  a  line  parallel  to  H,  V  the  vanishing-point, 
and  h  Vthe  trace  of  the  projecting  plane  on  the  picture- 
plane.  Then  /?  is  the  angle  the  visual  ray  makes  with 
H,  and  h  V  the  perpendicular  distance  from  the  horizon 
to  the  vanishing-point. 

h  V 
In  this  right  triangle  sin /?  =  ~-y,   or   &V  =  SVsin  /?. 

Substituting  the  value  of  S  V  from  (i)  in  this  equation, 
we  have 

&  V  =  SV  D^cosecasin/9 (3) 

Equation  (3)  gives  the  distance,  h  V,  that  the  vanish- 
ing-point is  above  or  below  the  horizon.  Now  to  find  the 
distance  to  the  right  or  left  of  the  vertical  line  through 
Sv  a  right  triangle  may  be  formed  lying  in  the  picture- 
plane  (V)  of  which  Ss  V  in  (2)  is  the  hypothenuse,  h  V  in 


126 


MECHANICAL   DRAWING. 


(3)   one  leg,   and  the  other  leg  the  required  distance. 
Calling  d  the  required  distance,  we  have 


or  substituting  the  value  of  SB  V  in  (2)  and  h  V  in  (3), 
we  have 


=^Sv  DB2  cot2  a  -SB  DB2  cosec2  a  sin2  /? 
S.  D., 


-Vcos2a-sin2/?. 


(4) 


sin  a 
If  /?  =  o°,  then  (3)  becomes  h  V  =  o, 

and  (4)  becomes      d  =  Sv  DB  cot  a, 

that  ,is,  the  perspectives  of  all  lines  lying  in  horizontal 
planes,  whatever  angle  they  make  with  V,  will  vanish 
in  the  horizon.  If,  however,  a=o°,  the  vanishing-point 
will  be  in  the  horizon  at  infinity. 

If  a  =  o°,  then  (3)  becomes  h  V=  oo, 
and    (4)  becomes       d=&>, 

for  all  values  of  /?,  i.e.,  the  vanishing-point  of  all  lines 
parallel  to  the  picture-plane  is  at  infinity,  and  the  per- 
spectives of  the  parallel  lines  of  any  system  of  lines  paral- 
lel to  the  picture-plane  are  parallel. 

If  a  =90°,  then  (2)  becomes  S0  V  =  o,  i.e.,  the  vanishing- 
point  of  lines  perpendicular  to  the  picture-plane  (V)  is 
atS,. 

If  a  =  45°  and  ;9  =  o°,  (3)  becomes  h  V  =  o, 

and  (4)  becomes     ^  =  St)Du, 


i.e.,  the  vanishing-point  of  diagonals  is  at  the  distance- 
point. 

26.  To  Find  the  Vanishing-point  of  any  System  of  Par- 
allel Lines,  Graphically.  In  Fig.  5,  let  H  V  be  the  line  of 
intersection  of  H  and  V,  SB  D,,  the  horizon  and  Sk  Ak 


FIG.  s- 


—  SB  AB  the  H  and  V  projections  of  a  line   through    the 
point  of  sight,  parallel  to  a  system  of  paralell  lines.     By 


GENERAL   INSTRUCTIONS   AND   CONVENTIONS. 


127 


Art.  23,  where  the  line  S  A  pierces  the  picture-plane  is 
the  vanishing-point,  v,  of  the  perspectives  of  these  lines. 

By  revolving  the  line  S  A  about  the  V  trace  of  its  V 
projecting  plane,  the  angle  a  which  the  line  makes  with 
the  picture-plane  may  be  found. 

Lay  off  from  Se  D,,,  at  Dr,  the  angle  <?,  equal  to  the 
complement  of  a,  and  find  where  D^  k  meets  Sv  S;,. 
With  S,,  k  as  a  radius  and  S,,  as  a  center  draw  the  arc 
k  v,  meeting  S0  \  at  v,  the  required  vanishing-point. 
The  triangles  Sr  Sr  v  and  Sv  Dv  k  are  equal  right  tri- 
angles; therefore  Svv  =  Sv  k. 


27.  When  the  horizontal  projection  of  the  point  of 
sight,  SA,  is  not  used,  as  is  usually  the  practice  in 
drawing  perspectives,  the  vanishing-point  of  any  system 
of  parallel  lines  may  be  found  by  the  following 

Rule :  Form  a  right  triangle  on  Sv  Dv  as  one  leg,  lay  off 
from  Sr  De  at  Dv  the  complement  of  the  angle  that  the  lines 
make  with  V  to  obtain  the  direction  of  the  hypotenuse. 
With  the  other  leg  as  a  radius  (Sv  k)  and  Sv  as  a  center 
describe  an  arc;  where  this  arc  meets  the  V  projection  of 
the  line  through  the  center  of  the  picture  will  be  the  required 
vanishing-point. 


128 


MECHANICAL  DRAWING. 


28. 


PLATE  XIX. 


Prob.  33.  Draw  the  perspective  of  a  rectangular,  trun- 
cated prism,  with  parallel  lines,  ]-"  apart,  on 
the  lateral  faces,  parallel  to  the  truncated 
base. 

The  base  of  the  prism  is  2''  by  I  \";  the  height 
of  the  prism  is  3".  The  wider  faces  make  an 
angle  of  30  degrees  with  V,  inclining  away 
from  it  to  the  right.  The  bottom  face  rests 
on  a  horizontal  plane  at  y  =  i".  The  upper 
face  is  partly  truncated  by  a  plane  perpen- 
dicular to  V  and  making  an  angle  with  H  of 
15  degrees,  inclining  downwards  to  the  right, 
beginning  on  the  top  base  at  a  point  in  the 
middle  of  the  shorter  side. 

One  edge  is  in  the  picture-plane  at  #  =  5|-" '. 

The  trace  on  H  of  the  picture-plane  (H  V)  is 
at  y  =  5i". 

The  horizon  is  at  y  =  4i". 

S0  is  at  x  =  &",  and  D,  is  at  x  =  g$". 

Next  find  the  vanishing-point  of  the  longer  side 
of  the  lower  base,  A'  B'  and  P  E'  and  the 
small  portion  of  the  upper  base  parallel  to 
it.  Since  these  are  horizontal  lines  their  per- 
spectives will  vanish  in  the  horizon.  The 
angle  they  make  with  V  is  30  degrees.  With 
S,,  Dv  as  one  leg  of  a  right  triangle  and  an 
angle  of  60  degrees  at  Dr  construct  the  tri- 
angle S0  Dr  w'  (w'  comes  off  the  plate),  then 
with  Sv  w'  as  a  radius  and  S,,  as  a  center 
describe  an  arc  cutting  the  horizon  in  v,  the 


required  vanishing-point.  (The  V  projec- 
tion of  the  line  through  Sr  parallel  to  the 
V  projection  of  A'  B'  coincides  with  the  ho- 
rizon.) In  a  similar  way  v'  is  found,  which 
is  the  vanishing-point  of  the  perspectives  of 
the  side  of  the  base  A'  F'. 

To  find  the  vanishing-point  of  the  perspectives 
of  the  sides  of  the  base  lying  in  the  truncated 
plane,  as  A  B  and  K  E:  First  determine  the 
angle  these  lines  make  with  V,  by  the  method 
of  determining  the  angle  a  line  makes 
with  a  plane  of  projection  as  shown  in  the 
triangle,  a  Bv  n.  Also  find  its  complement, 
p  n  B^.  Through  Sv  draw  a  line,  SvVi, 
parallel  to  the  vertical  projection  of  A  B, 
which  is  the  line  a  Bv.  With  S^  DB  as  one 
leg  of  a  right  triangle  and  the  angle  at  Dt 
equal  to  p  n  Bv  construct  the  triangle 
S^  Dv  w';  then  with  Sv  w'  as  a  radius  and 
S^  as  a  center  describe  the  arc  cutting  the 
line  Sv  Vi  in  v\,  the  required  vanishing-point. 
Similarly  a/,  the  vanishing-point  of  the  per- 
spectives of  E  B  and  A  G,  may  be  found. 
The  perspective  of  F  G  vanishes  at  v',  and 
that  of  F  K  at  v.  The  perspectives  of  the 
parallel  lines  on  the  face  A  B  A'  B',  which  are 
parallel  to  the  truncated  face,  vanish  at  vit 
and  those  on  the  face  A  F  A'  F'  vanish  at 
Vir.  The  section-lines  on  the  truncated  base 
are  perpendicular  to  V  and  vanish  at  S,. 


PLATE  XIX. 


129 


130 


MECHANICAL   DRAWING. 


29. 


PLATE  XX. 


Prob.  34.  Find  the  perspective  of  the  body  in  Prob.  32, 
Plate  XVIII,  and  the  perspectives  of  its 
shades  and  shadows. 

LetH  Vbeaty  =  5". 

The  horizon  at  y  =  4". 

S^  at  x  =  7",  and 

D,at*-2". 

The  front  face  of  the  base  is  taken  in  the  pic- 
ture-plane, the  object  being  in  the  third 
angle. 

Use  the  tracing  made  of  the  problem  referred 
to  instead  of  drawing  again  the  projec- 
tions. 

Find  the  vanishing-point  of  the  perspectives  of 
the  shadows  of  vertical  edges;  it  should  fall 
at  D... 


Where  there  are  only  a  few  lines  that  are 
parallel  in  a  perspective,  it  is  often  simpler  to 
find  the  perspectives  of  the  extremities  of  the 
lines  than  to  attempt  to  find  the  vanishing- 
point  of  their  perspectives. 

In  this  problem,  the  shadow  having  already 
been  found,  the  perspective  of  the  shadow 
may  be  determined  directly  by  finding  the 
perspectives  of  the  outline,  but  in  general 
the  perspective  of  the  shadow  is  found  with- 
out first  finding  the  shadow,  but  rather  by 
finding  where  the  perspective  of  a  ray  of 
light  through  a  point  meets  the  perspective 
of  its  projection  on  the  plane  or  surface  upon 
which  the  shadow  of  the  point  s  cast  (see 
the  method  of  Part  I). 


30. 


PLATF  XXI. 


Prob.  35.  Find  the  perspective  of  the  same  body,  sub- 
stituting a  cylinder  for  the  hexagonal  pillar. 
The  radius  of  the  base  to  be  f-". 


Other  problems  in  perspective  may  be  added  by  using 
the  problems  in  Part  I. 


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